Mercury and its toxic effects on fish

1.   Introduction

Dispersal of organisms is a topic of central interest in ecology and evolutionary biology. Its effects on the size, stability, and interactions of populations, as well as biological invasions and the geographical distribution of populations have attracted considerable studies. Investigation on dispersal strategies which are evolutionarily stable has been the fundamental research goal for theoretical ecologists ^[7,26]. The relationship between diffusion rates, spatial heterogeneity, and coupling from competition of species is the target of several recent works. To tackle these problems, continuous diffusion models expressed by reaction-diffusion systems have been considered in ^{[1,2,3,6,9,13,14,19,20,23,25]}. On the other hand, discrete diffusion models represented by systems of ODEs have been investigated in ^{[4,5,11,12,24,31]}.

Concerning the interaction between diffusion rates and the heterogeneity of the environment and mutant invasion, the following competitive Lotka-Volterra model was investigated in ^[18,19]:

under homogeneous Neumann boundary condition, where $\mu$ is the diffusion rate and functions $\alpha(x)$ and $\beta(x)$ express the spatially dependent intrinsic growth rates or reproductive rates of $u$- and $v$-species, respectively. Therein, to study the effect of spatially heterogeneous growth rates on the competitive dynamics, the difference between intrinsic growth rates of two species was set as

where $g(x)$ is a function describing resource difference between two species from the viewpoint of spatial heterogeneity, and $\tau > 0$ measures the magnitude of the difference. The case $g(x) > 0$ on the considered domain was studied in ^[18], whereas the situation that $g(x)$ changes sign was investigated in ^[19]. The assumption in ^[19],

means that the mutant $u$-species has better average reproductive rate than $v$-species, and thus the total population of $u$-species has higher growth rate than that of $v$-species when two populations are identical in the whole space $\Omega$. However, under such a circumstance, $u$-species possibly fails to invade when rare for certain level of diffusion rate. Mathematically, stability of semitrivial solutions $(\tilde{u}, 0)$ and $(0, \tilde{v})$, which depend on the magnitudes of $\mu$ and $\tau$, was analyzed in ^[19]. The stability may switch according to the varying diffusion rate $\mu$. In particular, by measuring the level of mutation with the value of $\tau$, theoretical analysis for the cases of tiny and large mutation was established therein. In the former case ($0 < \tau\ll1$), multiple switches of global convergence to different equilibria was derived and the relationship between the bifurcation value of the diffusion rate and the value of $\tau$ was also established, while in the latter case ($\tau\gg1$), only once switch of global convergence was observed.

The influence from magnitudes of diffusion rates on the competition outcome has been another topic of interest. It has been shown in ^[9] that the slower diffuser always prevails if the two species interact identically with the environment, see also ^[12,20,23]. To focus on the effect of diffusion rates, the birth rates for all competing species were set equal to the carrying capacity of the environment, see ^{[2,5,6,13,15]}.

Models for competitive species with dispersal expressed by discrete diffusion are also very appealing. Indeed, organisms are distributed in space, often in patches of habitat scattered over a landscape and region, and the distribution is determined by the pattern of movement between these patches. More specifically, it is interesting to see how possible interaction outcome, which can be competitive exclusion and coexistence, depends on the diffusion rates and the birth rates.

As early as in 1934, Gause ^[10] formulated the competitive exclusion law which in particular states that the species with a larger birth rate will outcompete the other one, if the other properties are the same. Concerning these issues, Gourley and Kuang ^[11] then asked how does diffusion affect the competition outcomes of two competing species that are identical in all respects other than their strategies on how they spatially distribute their birth rates. They studied the following ODE system as a model for two neutrally competing species on two patches of habitat:

where $u_{i}$ (resp., $v_{i}$) is the population density of species-$u$ (resp., -$v$) in patch $i$, $i = 1, 2$; the linear birth rates $\alpha_1, \alpha_2, \beta_1, \beta_2$ are positive parameters, and there is a diffusion between the two patches with same diffusivity (dispersal rate) $d$ for both species. The two species differ only in their birth rates. Let $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ denote the semitrivial equilibrium with extinct $v$-species. The following conjectures on the global dynamics of system (1.3) were posed in ^[11]:

Conjecture 1. Assume that in system (1.3), $\beta_{1}- \sigma = \alpha_{1} < \beta_{1} < \beta_{2} < \alpha_{2} = \beta_{2}+ \sigma$ with $0 < \sigma < \beta_{1}$, and $d$ is sufficiently large. If $u_{1}(0)+u_{2}(0) > 0$, then

Conjecture 2. Assume that in system (1.3), $\beta_{1}- \sigma = \alpha_{1} < \beta_{1} < \beta_{2} < \alpha_{2} = \beta_{2}+ \sigma$ with $0 < \sigma < \beta_{1}$, and $d$ is small enough so that (1.3) has a positive steady state $e_{*}$. If $u_{1}(0)+u_{2}(0) > 0$ and $v_{1}(0)+v_{2}(0) > 0$, then

These conjectures, if true, suggest that the species that can concentrate its birth in a single patch wins, if the diffusion rate is larger than a critical value. That is, the winning strategy is to focus as much birth in a single patch as possible. In ^[24], the following global dynamics and bifurcation were established, which include confirmation of Conjectures 1 and 2:

Theorem 1.1. Suppose that the following condition holds in system (1.3),

Then there is a constant $\tilde{d} > 0$ which can be expressed or estimated by the birth rates, so that if $d \ge \tilde{d}$, $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is globally asymptotically stable among the initial data in $\mathbb{R}_{+}^{4}$ satisfying $u_{1}(0)+u_{2}(0) > 0$; if $d < \tilde{d}$, (1.3) has a unique positive steady state $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ which is globally asymptotically stable among the initial data in $\mathbb{R}_{+}^{4}$ satisfying $u_{1}(0)+u_{2}(0) > 0$ and $v_{1}(0)+v_{2}(0) > 0$.

Note that $\alpha_1+\alpha_2$ and $\beta_1+\beta_2$ measure the average birth rates of species $u$ and $v$, respectively. The condition of Theorem 1.1 means $\alpha_{1} < \beta_{1} < \beta_{2} < \alpha_{2}$ and $\beta_{1}+ \beta_{2} \leq \alpha_{1}+\alpha_{2}$, and indicates that the birth rate of $u$-species is larger than that of $v$-species in the second patch, and less than that of $v$-species in the first patch, whereas the average birth rate of $u$-species is larger than or equal to that of $v$-species. For the situation with identical average birth rate: $\alpha_1+\alpha_2 = \beta_1+\beta_2$, i.e., the case in these conjectures, Theorem 1.1 implicates that the two species coexist in a slow diffusion environment, whereas in a fast diffusion environment, the species that can concentrate its birth in a single patch drives the other species into extinction. Convincingly, the same scenario prevails when $u$ has further competitive advantage that its average birth rate is larger than $v$-species: $\alpha_1+\alpha_2 > \beta_1+\beta_2$. Along with such finding is that the semitrivial equilibrium $(0, 0, \bar{v}_1, \bar{v}_2)$ is always unstable for any diffusion rate $d$, as was stated in Proposition 3.11 of ^[24]. It becomes very interesting to see what happens when $\alpha_1+\alpha_2 < \beta_1+\beta_2$, i.e., $v$-species has larger average birth rate.

In this paper we will examine such interesting situation, i.e., system (1.3) under condition

This condition means $\alpha_{1} < \beta_{1} < \beta_{2} < \alpha_{2}$, and $\alpha_{1}+\alpha_{2} < \beta_{1}+ \beta_{2}$, due to $(\beta_{1}+ \beta_{2})-(\alpha_{1}+\alpha_{2}) = \sigma_{1}-\sigma_{2} > 0$. That is, the birth rate $\alpha_{2}$ of $u$-species in the second patch is the biggest among all species and patches, but the average birth rate of $v$-species is larger than that of $u$-species; one may also regard this as that $v$-species has more total resources than $u$-species. Then we ask how the magnitude of the dispersal rate $d$ is related to the species persistence or extinction. With the framework of monotone dynamics, we shall target the global dynamics of system (1.3) and the bifurcation with respect to $d$, under condition (C).

It turns out that the dynamical scenarios are richer than the case under condition (C$'$). In particular, equilibrium $(0, 0, \bar{v}_1, \bar{v}_2)$ switches from being unstable to stable, as $d$ increases. On the other hand, there are up to two stability changes for equilibrium $(\bar{u}_1, \bar{u}_2, 0, 0)$, as $d$ increases. That is, the property described as monotone relation between the stability of $(\bar{u}_1, \bar{u}_2, 0, 0)$ and the diffusion rate $d$ no longer holds, cf. ^[19]. The main results will be summarized in Theorem 4.2. There are two dynamical scenarios (see Figure 1): (ⅰ) Under $\sigma_2 \beta_2 < \sigma_1 \beta_1$, there exists an $d^{*}_{3} > 0$, so that the positive steady state $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ is globally attractive for $d < d^{*}_{3}$ and the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ becomes globally attractive for $d \geq d^{*}_{3}$. (ⅱ) Under $\sigma_2 \beta_2 > \sigma_1 \beta_1$, there exist $d^{*}_{1}, d^{*}_{2}, d^{*}_{3}$ with $0 < d^{*}_{1} < d^{*}_{2} < d^{*}_{3}$, so that $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ is globally attractive for $d < d^{*}_{1}$ or $d^{*}_{2} < d < d^{*}_{3}$, $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is globally attractive for $d^{*}_{1}\leq d \leq d^{*}_{2}$, and $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ becomes globally attractive for $d \geq d^{*}_{3}$. In addition, $d^{*}_{1}, d^{*}_{2}, d^{*}_{3}$ can be estimated in terms of the system parameters. Our analytical work on the model strongly suggests that, in a fast diffusion (large dispersal) environment, a species will prevail if its average birth rate is larger than the other competing species; in a slow diffusion (small dispersal) environment, the two species can coexist or one species that has the greatest birth rate among both species and patches, even with smaller average birth rate, will be able to persist and drive the other species to extinction.

We note that $\alpha_1+\alpha_2 > \beta_1+\beta_2$ in system (1.3) is analogous to condition (1.2) in PDE system (1.1). The present study, with $\alpha_1+\alpha_2 < \beta_1+\beta_2$, can be compared to the results in ^[19] with $u$ and $v$ reversed. Systems with two competing species over two patches with different dispersal rates and more general competition coupling have been considered in ^[21,29,30]. While the effect of competition was studied in ^[29], herein we aim at investigating the influence of both dispersal rate and birth rates on the population dynamics and assume the same ability of competition for two species in (1.3). Predator-prey dynamics on two-patch environments were investigated in ^[8,16,22].

This presentation is organized as follows. In Section 2, we characterize the existence of positive equilibrium for system (1.3). In Section 3, we analyze the stability of the semitrivial equilibria. In Section 4, we discuss the existence of positive steady state representing coexistence of two species and extinction of one species, depending on the magnitude of dispersal rate. Four numerical examples illustrating the present theory are given in Section 5. We summarize our results with some discussions in Sections 6. For reader's convenience, we review in Appendix Ⅰ the monotone dynamics theory which is to be applied to obtain our results. Some qualitative properties of the semitrivial equilibria for system (1.3) reported in ^[24] are recalled in Appendix Ⅱ.

2.   Existence of positive equilibrium

In this section, we characterize the conditions under which the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ of system (1.3) exists. There are five parameters $\alpha_1, \alpha_2, \beta_1, \beta_2, d$ in system (1.3), which generate a complication of analysis for such existence. We first derive the following magnitude relationships which are required in the main result, Theorem 2.1, of this section.

Lemma 2.1. The following parameter relationships hold under condition (C).

(ⅰ) $\frac{1}{\sigma_{1}-\sigma_{2}} < \frac{\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$ if and only if $(\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2)(\sigma_1\beta_{1}-\sigma_2\beta_{2}) > 0$.

(ⅱ) $0 < \frac{\alpha_{1}\alpha_{2}}{\sigma_2\alpha_{2}^2-\sigma_1\alpha_{1}^2} < \frac{\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$, provided $\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2 > 0$.

(ⅲ) $\frac{\sigma_{1}}{\sigma_{1}-\sigma_{2}} < \frac{\beta_{2}}{\sigma_{1}+\sigma_{2}}$, provided $\sigma_1\beta_{1} > \sigma_2\beta_{2}$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$.

Proof. Recall that $\sigma_1 > \sigma_2$ in condition (C).

(ⅰ) We compute

Thus, $\frac{\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}-\frac{1}{\sigma_{1}-\sigma_{2}} > 0$ if and only if $(\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2)(\sigma_1\beta_{1}-\sigma_2\beta_{2}) > 0$.

(ⅱ) Suppose $\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2 > 0$. Then

The assertion follows from

due to $\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1} = \alpha_{1}\beta_{2}-(\beta_2+\sigma_2)(\alpha_1+\sigma_1) < 0$.

(ⅲ) If $\sigma_1\beta_1 > \sigma_2 \beta_2$, then

provided $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$. The assertion thus follows.

The following parameter condition is to be used throughout the discussions:

Certainly condition $({\rm P})$ holds only if $\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2 > 0$. And a direct computation shows that condition $({\rm P})$ is equivalent to $\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2 > 0$ with

Accordingly, if condition $({\rm P})$ holds, Lemma 2.1(ⅰ) can be recast as

for convenience of later use, we put this relationship as

The condition of Lemma 2.1(ⅲ): $\sigma_1\beta_1 > \sigma_2 \beta_2$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$ implies

Then, by combining condition $({\rm P})$, we obtain

On the other hand, combining $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} > \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$, i.e. $\sigma_2 \beta_2 > \sigma_1\beta_1$ by (2.2), with condition $({\rm P})$ yields

Therefore, by imposing condition (P) additionally, the following relationships can be concluded.

Lemma 2.2. Assume that conditions (C) and $({\rm P})$ hold.

(ⅰ) If $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$, then

(ⅱ) If $\sigma_2\beta_{2} > \sigma_1\beta_{1}$, then

It is obvious that the terms in the inequalities in Lemma 2.2 can be simplified. But it is convenient to keep these forms.

Remark 1. In Lemma 2.2, with (2.1), the condition in (ⅱ) : $\sigma_2\beta_{2} > \sigma_1\beta_{1}$ leads to $\sigma_1 \beta_1 \beta_2 < \sigma_2 \beta^{2}_2 < \sigma_1 \beta_1(\beta_1+\sigma_1+\sigma_2)$, and thus $\beta_2- \beta_1 < \sigma_1+\sigma_2$, which is contrary to the condition $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$ in (ⅱ). That is, the condition in (ⅰ) and the condition in (ⅱ) are opposite cases under assumption $\rm (P)$. In addition, the condition in Lemma 2.2(ⅰ): $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$ further indicates

via (2.1).

We characterize the existence of positive equilibrium for system (1.3) in the following theorem.

Theorem 2.1. Consider system (1.3) under conditions (C) and $\rm (P)$.

(ⅰ) Under $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$, there exists an $d^{*}_{3} > 0$ so that the system has a unique positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ if and only if $0 < d < d^{*}_{3}$.

(ⅱ) Under $\sigma_2\beta_{2} > \sigma_1\beta_{1}$, there exist $d^{*}_{1}, d^{*}_{2}, d^{*}_{3} > 0$, with $d^{*}_{1} < d^{*}_{2} < d^{*}_{3}$, so that the system has a unique positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ if and only if $0 < d < d^{*}_{1}$ or $d^{*}_{2} < d < d^{*}_{3}$.

In addition,

Proof. System (1.3) has a positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ if and only if

are satisfied for $u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2} > 0$. Let $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ be a solution of (2.6) and denote

Combining each pair of equations in (2.6), we obtain

This yields

and $k > 1$, as $0 < \sigma_2 < \sigma_1$. Substituting $b = k/a$ and $a = k/b$ into (2.8) respectively leads to

We thus express $a, b$ in terms of system parameters, and it can be computed that $b^{2} < k < a^{2}$ and $a > 1$. We substitute (2.7) into (2.6) and obtain

Solving the two equations in (2.11), we have

On the other hand, solving the two equations in (2.12), we obtain

In fact, (2.13) and (2.14) are equivalent, as it can be seen by (2.8) that $\alpha_{1}+ad-d = \beta_{1}+bd-d$ and

Herein, $\alpha_{1}+ad-d > 0$ since $a > 1$. From (2.13) and (2.14), we obtain

Observe that $u^{*}_{1}, v^{*}_{1} > 0$ imply $u^{*}_{2}, v^{*}_{2} > 0$, due to (2.7) and $a, b > 0$. Hence, system (1.3) has a unique positive equilibrium if and only if

and

To explore the range of $d$ where $u^{*}_{1}, v^{*}_{1} > 0$, we denote $u^{*}_{1}(d), v^{*}_{1}(d)$ to express the dependence of $u^{*}_{1}, v^{*}_{1}$ on $d$. Let us discuss the positivity of $v^{*}_{1}$ first. The terms in the brackets of (2.16) can be recast as

We define two functions to discuss the positivity of $v^{*}_{1}$:

It follows from (2.16) and (2.17) that

In addition, $v^{*}_{1} > 0$ if and only if $F_{v^{*}}(d) > 0$ if and only if $G_{v^{*}}(d) > 0$, due to $b^2 < k < a^2$. In the following discussions (a)-(e), we analyze the ranges of $d$ within which $F_{v^{*}}(d)$ and $G_{v^{*}}(d)$ take positive or negative values. For some situations, analyzing $F_{v^{*}}$ is more convenient than $G_{v^{*}}$, whereas the convenience is reverse in other cases.

(a) $F_{v^{*}}(d) > 0$ if $d < \frac{\sigma_1 \sigma_2\alpha_{1}\alpha_{2}}{\sigma_2\alpha_{2}^2-\sigma_1\alpha_{1}^2}$: It can be seen that $a\alpha_{1} > \alpha_{2}$ implies $F_{v^{*}}(d) > 0$, due to $a > 1$. On the other hand, $a\alpha_{1} > \alpha_{2}$ is actually

which is equivalent to $d < \frac{\sigma_1 \sigma_2\alpha_{1}\alpha_{2}}{\sigma_2\alpha_{2}^2-\sigma_1\alpha_{1}^2}$.

(b) $F_{v^{*}}(d) > 0$ if $d > \frac{\sigma_{1}\sqrt{\sigma_{1}\sigma_{2}}+\sigma_{1}\sigma_{2}}{\sigma_{1}^{2}-\sigma_{2}^{2}}(\alpha_{2}-\alpha_{1})$: If $b > 1$, i.e. $d > \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$ by (2.10), then $a < k$ since $ab = k$. From $b^{2} < k < a^{2}$, we have $1 < b < \sqrt{k} < a < k$. Then

if $d > \frac{k}{(k+1)(\sqrt{k}-1)}(\alpha_{2}-\alpha_{1}) = \frac{\sigma_{1}\sqrt{\sigma_{1}\sigma_{2}}+\sigma_{1}\sigma_{2}}{\sigma_{1}^{2}-\sigma_{2}^{2}}(\alpha_{2}-\alpha_{1})$. It is clear that

(c) $G_{v^{*}}(d) > 0$ if $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} < d < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$: Obviously, $G_{v^{*}}(d) > 0$ if $k\beta_{1}-b\beta_{2} > 0$ and $b > 1$, which is

This is equivalent to

by (2.10). Such value of $d$ exists provided $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$.

(d) $G_{v^{*}}(d) < 0$ if $\frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} < d < \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$: $G_{v^{*}}(d) < 0$ provided $k\beta_{1}-b\beta_{2} < 0$ and $b < 1$, which is

and equivalent to $\frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} < d < \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$, provided $\frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} < \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$.

(e) $G'_{v^{*}}(d) < 0$ if $d < \min\left\{\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}, \frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}}\right\}$ and $G'_{v^{*}}(d) > 0$ if $d > \max\left\{\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}, \frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}}\right\}$: From (2.10), we compute

and $G'_{v^{*}}(d) = (k+1)(b-1)+b'(kd-\beta_{2}+d)$. It can be seen that $G'_{v^{*}}(d) < 0$, provided $b < 1$ and $kd-\beta_{2}+d < 0$, which are equivalent to $d < \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$ and $d < \frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}}$. On the contrary, $G'_{v^{*}}(d) > 0$, if $b > 1$ and $kd-\beta_{2}+d > 0$, which are equivalent to $d > \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$ and $d > \frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}}$.

For case (ⅰ), we will show that $v^{*}_{1}(d) > 0$ for all $d > 0$ if $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$. Recall Lemma 2.2(ⅰ): $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} < \frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$. From the above (c), (e), we summarize

In addition, at $d = \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$, i.e., $b = 1$, we have $G_{v^{*}}(\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}) = k\beta_1-\beta_2 > 0$, thanks to $\sigma_2\beta_{2} < \sigma_1\beta_{1}$. Therefore, from (2.19), we see that $G_{v^{*}}(d) > 0$ for all $d > 0$, namely, $v^{*}_{1}(d) > 0$ for all $d > 0$, under $\sigma_2\beta_{2} < \sigma_1\beta_{1}$, $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$, and condition (${\rm P}$).

For case (ⅱ), if $\sigma_2\beta_{2} > \sigma_1\beta_{1}$, we will show that $G_{v^{*}}(d) > 0$, and hence $v^{*}_{1}(d) > 0$, for $d$ in certain range. Recall Lemma 2.2(ⅱ): $\frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} < \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$. With the above (a), (b), (d), (e), we obtain

and

Furthermore, if $d\geq \frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}}$, i.e. $d(k+1)\geq \beta_{2}$, and $d\leq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$, i.e. $b\leq1$, we have

by $\sigma_2\beta_{2} > \sigma_1\beta_{1}$. To summarize, $G_{v^{*}}(d) < 0$ if $\frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} \leq d \leq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$. Therefore, from (2.20) and (2.21), there exists a unique $d^{*}_{1} > 0$ so that $G_{v^{*}}(d) > 0$ if $d < d^{*}_{1}$ and $G_{v^{*}}(d_1^*) = 0$, where

and there exists a unique $d^{*}_{2} > 0$ so that $G_{v^{*}}(d) > 0$ if $d > d^{*}_{2}$, where

The two cases for the assertion of $v^{*}_{1}(d) > 0$ are thus concluded. Now let us discuss the positivity of $u^{*}_{1}(d)$. From (2.15), we have

Let

Then

In addition, $u^{*}_{1} > 0$ if and only if $F_{u^{*}}(d) > 0$ if and only if $G_{u^{*}}(d) > 0$, due to $b^2 < k < a^2$. Let us discuss the signs of $F_{u^{*}}(d)$ and $G_{u^{*}}(d)$ in the following (a')-(d').

(a') $G_{u^{*}}(d) > 0$ if $d\leq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$: It is clear that $b\leq 1$ implies $G_{u^{*}}(d) > 0$, and $b\leq 1$ is equivalent to $d\leq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$. Thus, $G_{u^{*}}(d) > 0$ if $d\leq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$.

(b') $F_{u^{*}}(d) < 0$ if $d > \frac{\sigma_{1}}{\sigma_{1}-\sigma_{2}}(\alpha_{2}-\alpha_{1})$: If $b > 1$, then from $1 < b < \sqrt{k} < a < k$, we have

if $d > \frac{k}{k-1}(\alpha_{2}-\alpha_{1}) = \frac{\sigma_{1}}{\sigma_{1}-\sigma_{2}}(\alpha_{2}-\alpha_{1})$.

(c') Case (ⅰ): $\sigma_2\beta_2 < \sigma_1\beta_1$, i.e., $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} < \frac{\sigma_{1}\sigma_{2}\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$ by (2.2). We claim that $F'_{u^{*}}(d) < 0$ for $d\leq \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$ and $G'_{u^{*}}(d) < 0$ for $d > \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$. Notably,

and

by direct computations, where $a' = a'(d), b' = b'(d)$. For the first term of (2.23), we see that

In addition, from (2.10), we compute

Hence, in (2.23), we confirm $F'_{u^{*}}(d) < 0$ for $d\leq \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$, due to $a > 1$ and $a' < 0$. Next, we discuss the third term of $G'_{u^{*}}(d)$ in (2.24), and claim that

From (2.10) and $b' > 0$ shown in (2.18), a direct computation shows

For $d > \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$, we compute directly

Note that $\frac{\sigma_1\sigma_2}{\sigma_1-\sigma_2} < \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$, according to Lemma 2.2(ⅰ). As seen in (b) above, $b > 1$ is equivalent to $d > \frac{\sigma_1\sigma_2}{\sigma_1-\sigma_2}$. Thus, we see from (2.24) that $G'_{u^{*}}(d) < 0$ for $d > \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$, due to (2.26), $b > 1$, and $b' > 0$.

(d') Case (ⅱ): $\sigma_2\beta_2 > \sigma_1\beta_1$, i.e. $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} > \frac{\sigma_{1}\sigma_{2}\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$ by (2.2). We claim that the third term of (2.24): $d(b^{2}+1)+2b\beta_{1}-\beta_{2} > 0$ for $d\geq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$. If so, then it can be seen from (2.24) and $b' > 0$, that $G'_{u^{*}}(d) < 0$ for $d\geq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$ which is equivalent to $b\geq 1$. For $d\geq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$, we obtain

due to $\beta_1 > \sigma_1 > 0$ and $\beta_2- \beta_1 < \sigma_1+\sigma_2$, mentioned in Remark 1.

For case (ⅰ), we summarize properties (a')-(c'):

Recall Lemma 2.2(ⅰ): $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} < \frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2}$, and that $G_{u^{*}}(d)$ and $F_{u^{*}}(d)$ have identical sign. There are two possibilities:

(Ⅰ) $F_{u^{*}}(\frac{\sigma_2\beta_2}{\sigma_1+\sigma_2})\geq 0$, i.e., $G_{u^{*}}(\frac{\sigma_2\beta_2}{\sigma_1+\sigma_2})\geq 0$: As $G'_{u^{*}}(d) < 0$ for $d > \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$ and $F_{u^{*}}(d) < 0$ if $d > \frac{\sigma_{1}}{\sigma_{1}-\sigma_{2}}(\alpha_{2}-\alpha_{1})$, there exists a unique $d^{*}_{3} > 0$ such that $G_{u^{*}}(d^{*}_{3}) = 0$, where

(Ⅱ) $F_{u^{*}}(\frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}) < 0$, i.e., $G_{u^{*}}(\frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}) < 0$: As $F_{u^{*}}(d) > 0$ for $d \leq \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}}$, $F'_{u^{*}}(d) < 0$ for $d \leq \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$, and $G'_{u^{*}}(d) < 0$ for $d > \frac{\sigma_2\beta_2}{\sigma_1+\sigma_2}$, we confirm that there exists a unique $d^{*}_{3} > 0$ such that $G_{u^{*}}(d^{*}_{3}) = 0$, where

Both (Ⅰ) and (Ⅱ) indicate that there exists a unique $d^{*}_{3} > 0$ such that $G_{u^{*}}(d^{*}_{3}) = 0$, $G_{u^{*}}(d) > 0$ if $d < d^{*}_{3}$, and $G_{u^{*}}(d) < 0$ if $d > d^{*}_{3}$, where

For case (ⅱ), from the above (a'), (b'), and (d'), we summarize

We thus conclude that there exists a unique $d^{*}_{3} > 0$ such that $G_{u^{*}}(d^{*}_{3}) = 0$, $G_{u^{*}}(d) > 0$ if $d < d^{*}_{3}$, and $G_{u^{*}}(d) < 0$ if $d > d^{*}_{3}$, where

From (2.13), we see that $v_1^*\rightarrow \alpha_1+ad-d > 0$ as $u_1^* \rightarrow 0^+$, i.e., $u_1^*$ and $v_1^*$ can not be zero simultaneously. From the above discussions, we confirm that $d^{*}_{2} < d^{*}_{3}$.

Combining the above discussions of two scenarios for $v_1^*(d) > 0$, and one single scenario for $u_1^*(d) > 0$, the assertions are thus justified, see Figure 2.

Remark 2. (Ⅰ) Under conditions (C) and $\rm (P)$, the proof of Theorem 2.1 actually indicate:

(ⅰ) If $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$, then $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})\rightarrow (0, 0, \bar{v}_{1}, \bar{v}_{2})$, as $d\rightarrow (d^{*}_{3})^{-}$, i.e., the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ degenerates and merges into the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ at $d = d^{*}_{3}$.

(ⅱ) If $\sigma_2\beta_{2} > \sigma_1\beta_{1}$, then $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})\rightarrow (\bar{u}_{1}, \bar{u}_{2}, 0, 0)$, as $d\rightarrow (d^{*}_{1})^{-}$, i.e., the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ degenerates and merges into the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ at $d = d^{*}_{1}$; $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)\rightarrow (u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$, as $d\rightarrow (d^{*}_{2})^{+}$, i.e., the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ becomes the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ as the value of $d$ increases through $d^{*}_{2}$; $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})\rightarrow (0, 0, \bar{v}_{1}, \bar{v}_{2})$, as $d\rightarrow (d^{*}_{3})^{-}$, i.e., the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ again degenerates and merges into the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ at $d = d^{*}_{3}$.

(Ⅱ) In Theorem 2.1(ⅱ), combining $\sigma_2\beta_2 > \sigma_1\beta_1$ and condition $\rm (P)$ yields $\beta_2-\beta_1 < \sigma_1+\sigma_2$, which is contrary to condition $\beta_2-\beta_1\geq \sigma_1+\sigma_2$ in case (ⅰ), as mentioned in Remark 1.

(Ⅲ) Although the same symbol $d_3^*$ is used in Theorem 2.1 (ⅰ) and (ⅱ), they represent different values under assumptions in (ⅰ) and (ⅱ), respectively.

(Ⅳ) With the setting $a: = \frac{u_2^*}{u_1^*}, b: = \frac{v_2^*}{v_1^*}$, and subsequently $ab = \frac{\sigma_1}{\sigma_2} = : k$, we always have $b^2 < k < a^2$. Notably, in ^[24], $0 < k \leq 1$ under assumption $\sigma_1 \leq \sigma_2$, and hence $b < 1$. This is disparate from the situation in Theorem 2.1 that $k > 1$, and hence $a > 1$, due to $\sigma_1 > \sigma_2$.

3.   Stability analysis of semitrivial equilibria

In this section, we analyze the stability of the semitrivial equilibria for system (1.3). We denote by $(\bar{u}_1, \bar{u}_2, 0, 0)$ and $(0, 0, \bar{v}_1, \bar{v}_2)$ the semitrivial (boundary) equilibria for system (1.3), and by $\bar{u}_i(d)$ and $\bar{v}_i(d), i = 1, 2$, to express the dependence of $\bar{u}_i$ and $\bar{v}_i$ on $d$. In Appendix Ⅱ, we recall some properties of semitrivial equilibria of system (1.3) in Propositions 3.7-3.10 of ^[24], which are independent of the order between $\sigma_1$ and $\sigma_2$. Herein, we add the following additional properties for the semitrivial equilibria, which shall be employed to discuss the stability of semitrivial equilibria.

Proposition 3.1. (ⅰ) If $\alpha_{1} < \alpha_{2}$, then $\bar{u}'_1(d) > 0, \bar{u}'_2(d) < 0$, $\bar{u}''_{1}(d) < 0$, and $\bar{u}''_{2}(d) > 0$, for all $d > 0$.

(ⅱ) If $\beta_{1} < \beta_{2}$, then $\bar{v}'_1(d) > 0, \bar{v}'_2(d) < 0$, $\bar{v}''_{1}(d) < 0$, and $\bar{v}''_{2}(d) > 0$, for all $d > 0$.

Proof. (ⅰ) $(\bar{u}_1, \bar{u}_2, 0, 0)$ is an equilibrium of (1.3) if and only if $\bar{u}_1$ and $\bar{u}_2$ satisfy

Differentiating (3.1) with respect to $d$, we obtain

where $\bar{u}'_{i}, i = 1, 2$, represent the derivatives of $\bar{u}_{i}$ with respect to $d$. Thus,

Note that

by Proposition A.3 (in Appendix Ⅱ). Thus $\bar{u}'_{1} > 0$ and $\bar{u}'_{2} < 0$. More detailed descriptions for $\bar{u}_{1}$ and $\bar{u}_{2}$ can be found in Proposition A.5. We further differentiate (3.3) with respect to $d$, and obtain

Thus,

Let us focus on the numerators. For $\bar{u}''_{1}$, we have

due to $\bar{u}'_{1} > 0$, $\bar{u}'_{2} < 0$ for all $d > 0$, and Proposition A.3. Thus, $\bar{u}''_{1} < 0$. For $\bar{u}''_{2}$, with (3.5) and (3.6), we have

For the first two terms in the bracket,

by Proposition A.3. For the third term, using (3.1), we have

since $\bar{u}_1 < \bar{u}_2$. Thus, $\bar{u}''_{2} > 0$.

Part (ⅱ) can be obtained by arguments similar to those for (ⅰ), using

This completes the proof.

Propositions A.3-A.6, in Appendix Ⅱ, and Proposition 3.1 are independent of the order between $\sigma_1$ and $\sigma_2$. Some of the following properties for the semitrivial equilibria hold under $\sigma_2 < \sigma_1$. The following notations will be helpful to recognize various related quantities:

Proposition 3.2. Under conditions (C) and $\rm (P)$, the following relationships among parameters hold:

(Ⅰ) $\frac{\bar{u}_2}{\bar{u}_1}(d)$ is strictly decreasing with respect to $d$.

(Ⅱ) If $\frac{\bar{u}_2}{\bar{u}_1} = \frac{\sigma_1}{\sigma_2}$, then $d = d_{1}$.

(Ⅲ) If $\frac{\bar{u}_2}{\bar{u}_1} = \frac{\beta_2}{\beta_1}$, then $d = d_{3}$.

(Ⅳ) If $\frac{\bar{u}_2}{\bar{u}_1} = \frac{\sqrt{\sigma_1}}{\sqrt{\sigma_2}}$, then $d = d_{5}$.

(Ⅴ) (ⅰ) If $\sigma_2\beta_2 < \sigma_1\beta_1$, then $\frac{\sigma_1}{\sigma_2} > \frac{\beta_2}{\beta_1}$ and $d_{1} < d_{2} < d_{3} < d_{4}$.

(ⅱ) If $\sigma_2\beta_2 > \sigma_1\beta_1$, then $\frac{\sigma_1}{\sigma_2} < \frac{\beta_2}{\beta_1}$ and $d_{1} > d_{2} > d_{3} > d_{4}$.

(ⅲ) If $\sigma_2\beta_2 = \sigma_1\beta_1$, then $\frac{\sigma_1}{\sigma_2} = \frac{\beta_2}{\beta_1}$ and $d_{1} = d_{2} = d_{3} = d_{4}$.

Proof. (Ⅰ) The assertion follows from $\bar{u}'_1(d) > 0, \bar{u}'_2(d) < 0$, as in the proof of Proposition 3.1.

(Ⅱ) If $\frac{\bar{u}_2}{\bar{u}_1} = \frac{\sigma_1}{\sigma_2}$, with $\bar{u}_1$ and $\bar{u}_2$ satisfying (3.1), we have

By eliminating $\bar{u}_1$, we have

Then

due to $\alpha_2 = \beta_2+\sigma_2$ and $\alpha_1 = \beta_1-\sigma_1$.

Cases (Ⅲ) and (Ⅳ) can be obtained by arguments similar to those for (Ⅱ). Now let us prove (Ⅴ), and the assertions will be justified by the following (a)-(c):

(a) It is clear that

(b) We see that

as, by a direct calculation,

(c) It holds that

due to

This completes the proof.

In Appendix Ⅰ, we compute the Jacobian matrix for system (1.3). At $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$, the Jacobian matrix is

and at $(0, 0, \bar{v}_{1}, \bar{v}_{2})$, the Jacobian matrix is

First, let us focus on the stability of semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$, by calculating the eigenvalues of the following submatrices in (3.6):

Theorem 3.3. Assume that conditions (C) and $\rm (P)$ hold for system (1.3).

(ⅰ) If $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_{2}-\beta_{1}\geq \sigma_{1}+\sigma_{2}$, the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable for all $d > 0$.

(ⅱ) If $\sigma_2\beta_{2} > \sigma_1\beta_{1}$, there exist $\bar{d}_{1}, \bar{d}_{2} > 0$, with $\bar{d}_{1} < \bar{d}_{2}$, so that the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable when $d < \bar{d}_{1}$ or $d > \bar{d}_{2}$ and is asymptotically stable when $\bar{d}_{1} < d < \bar{d}_{2}$.

In addition,

Proof. Under condition (C), the two eigenvalues of the first matrix in (3.11) are negative by Gerschgorin's Theorem and Proposition A.3. Thus, the stability of $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is determined by the two eigenvalues, denoted by $\lambda_{\mp}$, of the second matrix in (3.11). By a direct calculation, the two eigenvalues are

First, we consider $\lambda_{-} = \lambda_{-}(d)$ and claim $\lambda_{-}(d) < 0$ for all $d > 0$. From condition (C) and Proposition A.3, we have

and

Then

We obtain

if $(\sigma_1+\sigma_2)+d \left(\frac{\bar{u}_1}{\bar{u}_2}-\frac{\bar{u}_2}{\bar{u}_1} \right)\geq 0$, and

if $(\sigma_1+\sigma_2)+d \left(\frac{\bar{u}_1}{\bar{u}_2}-\frac{\bar{u}_2}{\bar{u}_1} \right) < 0$. Consequently, $\lambda_{-}(d) < 0$ for all $d > 0$.

Next, we identify the sign of $\lambda_{+} = \lambda_{+}(d)$. Note that $\lambda_{+}(d)\geq 0$ if and only if

equivalently,

As $\beta_{1} = \alpha_{1}+\sigma_{1}$ and $\beta_{2} = \alpha_{2}-\sigma_{2}$, (3.12) can be expressed by

i.e.,

using (3.1). This inequality can be simplified to

From (3.10), we define

Then $\lambda_{+}(d)\geq 0$ if and only if $g(d)\leq 0$. According to Propositions A.3 and A.5, we have that $1 < \frac{\bar{u}_2(d)}{\bar{u}_1(d)} < \frac{\alpha_2}{\alpha_1}$ and $\frac{\bar{u}_2(d)}{\bar{u}_1(d)}$ decreases from $\frac{\alpha_2}{\alpha_1}$ to $1$ as $d$ increases from $0$ to $\infty$. Thus, $g(0) = -\sigma_1 \sigma_2$ and $g(d)\rightarrow -\infty$ as $d\rightarrow \infty$, because of $\sigma_1 > \sigma_2$. More precisely,

Note that $\frac{\bar{u}_2(d)}{\bar{u}_1(d)}\leq \frac{\sqrt{\sigma_1}}{\sqrt{\sigma_2}}$ is equivalent to $d\geq d_5$, by Proposition 3.2. Hence,

Thus,

A direct calculation yields

We know $\sigma_{2}(\frac{\bar{u}_2}{\bar{u}_1})'-\sigma_{1}(\frac{\bar{u}_1}{\bar{u}_2})' < 0$ for all $d > 0$, due to $\bar{u}'_1 > 0$ and $\bar{u}'_2 < 0$, as in the proof of Proposition 3.1 or by Proposition A.5; $\sigma_{2}\frac{\bar{u}_2}{\bar{u}_1}-\sigma_{1}\frac{\bar{u}_1}{\bar{u}_2} = \frac{\sigma_2 \alpha^{2}_2-\sigma_1 \alpha^{2}_1}{\alpha_1 \alpha_2} > 0$ when $d = 0$, by Lemma 2.1(ⅱ); $\sigma_{2}\frac{\bar{u}_2}{\bar{u}_1}-\sigma_{1}\frac{\bar{u}_1}{\bar{u}_2}\rightarrow \sigma_2-\sigma_1 < 0$ as $d\rightarrow +\infty$, due to Propositions A.3 and A.5. That is, $\sigma_{2}\frac{\bar{u}_2}{\bar{u}_1}-\sigma_{1}\frac{\bar{u}_1}{\bar{u}_2}$ decreases from $\frac{\sigma_2 \alpha^{2}_2-\sigma_1 \alpha^{2}_1}{\alpha_1 \alpha_2}$ to $-(\sigma_1-\sigma_2)$ as $d$ increases from $0$ to $+\infty$. On the other hand, by (3.1), we have

With (3.15), we reexpress (3.14) as

It follows that

by Proposition 3.1. Thus, the graph of $g(d)$ is concave downward. Therefore, there are two possible situations based on the above analysis: (ⅰ) $g(d) < 0$ for all $d > 0$, (ⅱ) there exist $\bar{d}_{1}, \bar{d}_{2} > 0$ such that $g(\bar{d}_{1}) = g(\bar{d}_{2}) = 0$, and

The graphs of $g(d)$ are illustrated in Figure 3. Accordingly, there are two possibilities for $\lambda_{+}$: (ⅰ) $\lambda_{+} > 0$ for all $d > 0$, (ⅱ) there exist $\bar{d}_{1}, \bar{d}_{2} > 0$ such that $\lambda_+ = 0$ for $d = \bar{d}_{1}, \bar{d}_{2}$ and

Now we investigate the two situations by analyzing $g(d)$ and the stationary equation (3.1). To determine the behavior of $g$, we seek for its equivalent expression. Let $w: = \frac{\bar{u}_2}{\bar{u}_1}$. As $\frac{\bar{u}_2}{\bar{u}_1}(d)$ is strictly decreasing with respect to $d$, by Proposition 3.2(Ⅰ), the one-to-one correspondence between $d$ and $w$ can be derived from the stationary equation for $\bar{u}_1$ and $\bar{u}_2$ in (3.1):

where $1 < w < \frac{\alpha_2}{\alpha_1}$, by Proposition A.5. Then

Let us define $q(w): = (\alpha_2-w\alpha_1)(\sigma_2 w^{2}-\sigma_1)-\sigma_{1}\sigma_{2}(w-1)(w^{2}+1)$, which is the numerator of $f(w)$, and thus $f(w) = \frac{q(w)}{(w-1)(w^{2}+1)}$. Note that

by $\beta_{1} = \alpha_{1}+\sigma_{1}$ and $\beta_{2} = \alpha_{2}-\sigma_{2}$. Thus, we have

due to $(w-1)(w^{2}+1) > 0$. In addition,

where

Notice that

according to Proposition 3.2(Ⅰ). By a direct computation in (3.18), we obtain

and

Case (ⅰ) $\sigma_2\beta_2 < \sigma_1\beta_1$, i.e., $\frac{\beta_2}{\beta_1} < \frac{\sigma_1}{\sigma_2}$: We first claim that $q(w) < 0$ for all $\frac{\beta_2}{\beta_1} < w < \frac{\sigma_1}{\sigma_2}$. Consider $w = \frac{\beta_2}{\beta_1}+\delta$, with $\delta > 0$ satisfying $\frac{\beta_2}{\beta_1} < w = \frac{\beta_2+\delta \beta_1}{\beta_1} < \frac{\sigma_1}{\sigma_2}$. Hence, $\sigma_2(\beta_2+\delta \beta_1) < \sigma_1\beta_1$. From (3.18), we obtain

by $\sigma_2\beta_2^{2}-\sigma_1\beta^{2}_1 > 0$ from condition $\rm (P)$, and $\sigma_1\beta_1 > \sigma_2(\beta_2+\delta \beta_1)$.

Next, we will show that $f'(\frac{\sigma_1}{\sigma_2}) < 0$ and $f'(\frac{\beta_2}{\beta_1}) > 0$. From (3.20) and (3.21), at $w = \frac{\sigma_1}{\sigma_2}$, a direct calculation yields

because of $\beta_2-\beta_1\geq \sigma_1+\sigma_2$. Thus, $f'(\frac{\sigma_1}{\sigma_2}) < 0$ by (3.20) and $(w-1)^{2}(w^{2}+1)^{2} > 0$. Similarly, from (3.20) and (3.21), at $w = \frac{\beta_2}{\beta_1}$, we have

due to condition $\rm (P)$, i.e., $\sigma_2 \beta^{2}_2 < \sigma_1 \beta_1(\beta_1+\sigma_1+\sigma_2)$ in (2.1). Thus, $f'(\frac{\beta_2}{\beta_1}) > 0$ by (3.20) and $(w-1)^{2}(w^{2}+1)^{2} > 0$.

Consequently, by (3.19) and concavity of $g(d)$, we conclude $f(w) < 0$ for all $1 < w = \frac{\bar{u}_2}{\bar{u}_1} < \frac{\alpha_2}{\alpha_2}$, i.e., $g(d) < 0$ for all $d > 0$, i.e., $\lambda_{+} > 0$ for all $d > 0$.

Case (ⅱ) $\sigma_2\beta_2 > \sigma_1\beta_1$, i.e., $\frac{\beta_2}{\beta_1} > \frac{\sigma_1}{\sigma_2}$: We claim that $q(w) > 0$ for all $\frac{\beta_2}{\beta_1} > w > \frac{\sigma_1}{\sigma_2}$. Consider $w = \frac{\sigma_1}{\sigma_2}+\delta$, with $\delta > 0$ satisfying $\frac{\beta_2}{\beta_1} > w = \frac{\sigma_1+\delta \sigma_2}{\sigma_2} > \frac{\sigma_1}{\sigma_2}$. Hence, $\sigma_2\beta_2 > (\sigma_1+\delta \sigma_2)\beta_1$. From (3.18), we compute

owing to $\sigma_1 > \sigma_2$ and $\sigma_2\beta_2 > (\sigma_1+\delta \sigma_2)\beta_1$. Combining (3.22) with (3.23), we have $q(w) > 0$ for all $\frac{\beta_2}{\beta_1}\geq w\geq\frac{\sigma_1}{\sigma_2}$. That is, $g(d) > 0$ if $d_{3} \leq d \leq d_{1}$, by Proposition 3.2. Recall the relationship between $d$ and $w(d) = \frac{\bar{u}_2(d)}{\bar{u}_1(d)}$ in Proposition 3.2. We will use the relationship to estimate $\bar{d}_{1}$ and $\bar{d}_{2}$. For $\frac{\bar{u}_2}{\bar{u}_1} > \frac{\beta_2}{\beta_1}$, we have $d < d_{3}$ by Proposition 3.2, and then

Thus, $g(d) > 0$ if $d_{4} \leq d < d_{3}$. According to (3.12), (3.13), and Proposition 3.2, we obtain

In addition, we recall Lemma 2.1(ⅱ): $0 < \frac{\sigma_1 \sigma_2 \alpha_1 \alpha_2}{\sigma_2 \alpha^{2}_2-\sigma_1 \alpha^{2}_1} < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} = d_{4}$. Accordingly, there exist $\bar{d}_{1}, \bar{d}_{2} > 0$ such that $\lambda_{+} = 0$ for $d = \bar{d}_{1}, \bar{d}_{2}$ and

where

By a direct computation, it can be seen that

Thus, for consistency with the ranges in the assertion of Theorem 2.1, we also write

This completes the proof.

Remark 3. Assume that conditions (C) and $\rm (P)$ hold. Under $\sigma_2\beta_2 = \sigma_1\beta_1$ and $\beta_2-\beta_1 = \sigma_1+\sigma_1$, we do have $g(d_{1} = d_{2} = d_{3} = d_{4}) = 0$ and $g'(d_{1} = d_{2} = d_{3} = d_{4}) = 0$, by Proposition 3.2(Ⅴ)(ⅲ) and the proof of Theorem 3.3, and hence $\bar{d}_{1} = \bar{d}_{2}$.

Next, let us focus on the boundary equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$. We shall discuss the stability of $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ through analyzing the eigenvalues of the following submatrices in (3.7):

Theorem 3.4. Consider system (1.3) under condition (C). There exists a $\bar{d}_{3} > 0$ so that the boundary equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is unstable if $d < \bar{d}_{3}$ and asymptotically stable if $d > \bar{d}_{3}$. In addition,

Proof. Under condition (C), it can be shown that the two eigenvalues of the second matrix in (3.24) are negative, by Gerschgorin's Theorem and Proposition A.4 in Appendix Ⅱ. The stability of $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is thus determined by the two eigenvalues, denoted by $\lambda_{\mp}$, of the first matrix in (3.24). By a direct calculation, these two eigenvalues are

From Proposition A.4, we have

and thus $\lambda_{-} < 0$ for all $d > 0$. Next, let us identify the sign of $\lambda_{+} = \lambda_{+}(d)$. Note that $\lambda_{+}(d)\geq 0$ if and only if

By $\alpha_{1} = \beta_{1}-\sigma_{1}$, $\alpha_{2} = \beta_{2}+\sigma_{2}$, (3.5), and Proposition A.4, (3.25) is equivalent to

This inequality can be simplified to

Now, we define

Then $\lambda_{+}(d)\geq 0$ if and only if $h(d)\leq 0$. By Propositions A.4 and A.6, we have $1 < \frac{\bar{v}_2}{\bar{v}_1}(d) < \frac{\beta_2}{\beta_1}$ and $\frac{\bar{v}_2}{\bar{v}_1}(d)$ decreases from $\frac{\beta_2}{\beta_1}$ to $1$, as $d$ increases from $0$ to $\infty$. Hence, $h(0) = -\sigma_{1}\sigma_{2} < 0$ and $h(d)\rightarrow \infty$ as $d\rightarrow \infty$ due to $\sigma_1 > \sigma_2$.

A direct calculation yields

Notably, $\sigma_{1}(\frac{\bar{v}_1}{\bar{v}_2})'-\sigma_{2}(\frac{\bar{v}_2}{\bar{v}_1})' > 0$ for all $d > 0$, $\sigma_{1}\frac{\bar{v}_1}{\bar{v}_2}-\sigma_{2}\frac{\bar{v}_2}{\bar{v}_1} = - \frac{\sigma_2 \beta^{2}_2-\sigma_1 \beta^{2}_1}{\beta_1\beta_2}$ if $d = 0$, and $\sigma_{1}\frac{\bar{v}_1}{\bar{v}_2}-\sigma_{2}\frac{\bar{v}_2}{\bar{v}_1}\rightarrow \sigma_1-\sigma_2 > 0$ as $d\rightarrow \infty$; namely, $\sigma_{1}\frac{\bar{v}_1}{\bar{v}_2}-\sigma_{2}\frac{\bar{v}_2}{\bar{v}_1}$ increases from $- \frac{\sigma_2 \beta^{2}_2-\sigma_1 \beta^{2}_1}{\beta_1\beta_2}$ to $\sigma_1-\sigma_2$ as $d$ increases from $0$ to $\infty$. Moreover, from (3.5), the equations for $\bar{v}_i$, we have

With (3.28) and (3.29), we obtain

and then

by Proposition 3.1. Thus, the graph of $h(d)$ is concave upward. Therefore, from the above discussions, there is a unique $\bar{d}_{3} > 0$ such that

and accordingly,

The graph of $h(d)$ is illustrated in Figure 4.

Now, we estimate the range for the values of $\bar{d}_{3}$. Function $h$ in (3.30) can be expressed by

via (3.5). Thus, inequality (3.26) is equivalent to

According to Proposition A.4(ⅰ), we have

Then,

and

Consequently, (3.31) holds with

This completes the proof.

4.   Coexistence of two species and extinction of one species

Let us summarize the main results in Sections 2 and 3:

(ⅰ) Under $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_2-\beta_1\geq \sigma_1+\sigma_2$, the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ exists if $d < d^{*}_{3}$ (Theorem 2.1); the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable for all $d > 0$ (Theorem 3.3) and the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is unstable if $d < \bar{d}_{3}$ and asymptotically stable if $d > \bar{d}_{3}$ (Theorem 3.4). Besides, the estimated range of $d^{*}_{3}$ in Theorem 2.1 coincides with the one of $\bar{d}_{3}$ in Theorem 3.4.

(ⅱ) Under $\sigma_2\beta_{2} > \sigma_1\beta_{1}$, the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ exists if $d < d^{*}_{1}$ or $d^{*}_{2} < d < d^{*}_{3}$ (Theorem 2.1); the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable if $d < \bar{d}_{1}$ or $d > \bar{d}_{2}$, and asymptotically stable if $\bar{d}_{1} < d < \bar{d}_{2}$ (Theorem 3.3); the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is unstable if $d < \bar{d}_{3}$ and asymptotically stable if $d > \bar{d}_{3}$ (Theorem 3.2). In addition, the estimated ranges of $d^{*}_{1}$ and $d^{*}_{2}$ in Theorem 2.1 respectively coincide with those of $\bar{d}_{1}$ and $\bar{d}_{2}$ in Theorem 3.3.

In fact, the following theorem reveals that these critical values of $d$ are consistent in determining the existence of the positive equilibrium and the stability of semitrivial equilibria, namely, $d^{*}_{1} = \bar{d}_{1}$, $d^{*}_{2} = \bar{d}_{2}$ and $d^{*}_{3} = \bar{d}_{3}$. Such interesting consistency makes precise the global dynamics of this competitive species model (1.3), under the framework of monotone dynamics. Let us elaborate.

Theorem 4.1. $d^{*}_{1} = \bar{d}_{1}$, $d^{*}_{2} = \bar{d}_{2}$ and $d^{*}_{3} = \bar{d}_{3}$.

Proof. From (3.26), we see that $h(d) = 0$ if and only if

That is, $\bar{d}_{3}$ satisfies

Let $\bar{b}: = \frac{\bar{v}_2}{\bar{v}_1}(\bar{d}_{3})$, then

Thus,

where $k = \frac{\sigma_1}{\sigma_2}$. Recall the definition of $b$ in (2.7). From Remark 2(Ⅰ), we see that $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})\rightarrow (0, 0, \bar{v}_{1}, \bar{v}_{2})$, as $d\rightarrow (d^{*}_{3})^{-}$. Therefore, recalling (2.10), we obtain

Noting that $b = b(d)$ in (2.10) is monotone in $d$ (shown in (2.18)), with (4.1), we thus conclude that $d^{*}_{3} = \bar{d}_{3}$.

If $\sigma_2\beta_{2} > \sigma_1\beta_{1}$, from (3.11), we have $g(\bar{d}_{1}) = 0$ and $g(\bar{d}_{2}) = 0$. Let $\bar{a}_1: = \frac{\bar{u}_2}{\bar{u}_1}(\bar{d}_{1})$, $\bar{a}_2: = \frac{\bar{u}_2}{\bar{u}_1}(\bar{d}_{2})$. Then

It follows from Remark 2(Ⅰ) that $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})\rightarrow (\bar{u}_{1}, \bar{u}_{2}, 0, 0)$, as $d \rightarrow (d^{*}_{1})^-$. Notice that $a = a(d)$ in (2.10) is monotone in $d$, shown in (2.25). Therefore, recalling (2.10), we have

With (4.2), we thus conclude that $d^{*}_{1} = \bar{d}_{1}$. In addition, by Remark 2(Ⅰ), we see that $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)\rightarrow (u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ as $d\rightarrow (d^{*}_{2})^{+}$. Consequently,

With (4.2), we conclude that $d^{*}_{2} = \bar{d}_{2}$. This completes the proof.

Combining the discussions in Sections 2 and 3 with the assertion in Theorem 4.1, we conclude that for system (1.3), either there exists a positive equilibrium representing the coexistence of two species or one species drives the other to extinction, depending on the magnitude of the dispersal rate $d$.

Theorem 4.2. Consider system (1.3) under conditions (C), and $({\rm P})$.

(Ⅰ) Assume that $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_2-\beta_1\geq \sigma_1+\sigma_2$ hold.

(ⅰ) If $d < d^{*}_{3}$, then the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ is stable, and

$\lim_{t \rightarrow \infty}(u_{1}(t), u_{2}(t)$, $v_{1}(t), v_{2}(t)) = (u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$, for all $(u_{1}(0), u_{2}(0), v_{1}(0), v_{2}(0)) \in {\mathbb R}^4_+$ with $u_{1}(0)+u_{2}(0) > 0$ and $v_{1}(0)+v_{2}(0) > 0$.

(ⅱ) If $d \geq d^{*}_{3}$, then $\lim_{t \rightarrow \infty}(u_{1}(t), u_{2}(t), v_{1}(t), v_{2}(t)) = (0, 0, \bar{v}_{1}, \bar{v}_{2})$, for all $(u_{1}(0), u_{2}(0)$, $v_{1}(0), v_{2}(0)) \in {\mathbb R}^4_+$ with $v_1(0) +v_2(0) > 0$.

(Ⅱ) Assume that $\sigma_2\beta_{2} > \sigma_1\beta_{1}$ holds.

(ⅰ) If $d < d^{*}_{1}$ or $d^{*}_{2} < d < d^{*}_{3}$, then the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ is stable, and $\lim_{t \rightarrow \infty}(u_{1}(t), u_{2}(t)$, $v_{1}(t), v_{2}(t)) = (u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$, for all $(u_{1}(0), u_{2}(0), v_{1}(0), v_{2}(0)) \in {\mathbb R}^4_+$ with $u_{1}(0)+u_{2}(0) > 0$ and $v_{1}(0)+v_{2}(0) > 0$.

(ⅱ) = If $d^{*}_{1}\leq d \leq d^{*}_{2}$, then $\lim_{t \rightarrow \infty}(u_{1}(t), u_{2}(t), v_{1}(t), v_{2}(t)) = (\bar{u}_{1}, \bar{u}_{2}, 0, 0)$, for all $(u_{1}(0), u_{2}(0)$, $v_{1}(0), v_{2}(0)) \in {\mathbb R}^4_+$ with $u_1(0) +u_2(0) > 0$.

(ⅲ) If $d \geq d^{*}_{3}$, then $\lim_{t \rightarrow \infty}(u_{1}(t), u_{2}(t), v_{1}(t), v_{2}(t)) = (0, 0, \bar{v}_{1}, \bar{v}_{2})$, for all $(u_{1}(0), u_{2}(0)$, $v_{1}(0), v_{2}(0)) \in {\mathbb R}^4_+$ with $v_1(0) +v_2(0) > 0$.

Proof. The assertions are based on the monotone dynamics theory which is reviewed in Appendix I. (I) Assume that $\sigma_2\beta_{2} < \sigma_1\beta_{1}$ and $\beta_2-\beta_1\geq \sigma_1+\sigma_2$ hold. (ⅰ) If $d < d^{*}_{3}$, then the positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ is unique, by Theorem 2.1, the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable, by Theorem 3.3, and the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is unstable, by Theorem 3.4. Therefore, the assertion follows from Theorem A.1. (ⅱ) If $d > d^{*}_{3}$, then case (a) of the trichotomy in Theorem A.2 does not hold, since the positive steady state does not exist, by Theorem 2.1; case (b) does not hold since $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable, by Theorem 3.3, and $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is asymptotically stable, by Theorem 3.4. Therefore, the assertion follows from case (c) of Theorem A.2.

(Ⅱ) Assume that $\sigma_2\beta_{2} > \sigma_1\beta_{1}$. (ⅰ) If $d < d^{*}_{1}$ or $d^{*}_{2} < d < d^{*}_{3}$, the argument is similar to the one in (Ⅰ)(ⅰ). (ⅱ) If $d^{*}_{1} < d < d^{*}_{2}$, then case (a) of the trichotomy in Theorem A.2 does not hold, since the positive equilibrium does not exist, by Theorem 2.1; case (c) does not hold since $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is unstable, by Theorem 3.2, and $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is asymptotically stable, by Theorem 3.1. Therefore, the assertion follows from case (b) of Theorem A.2. (ⅲ) If $d > d^{*}_{3}$, the argument is similar to the one in (Ⅰ)(ⅱ). This completes the proof.

Remark 4 That the equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is globally asymptotically stable for $d > d^{*}_{3}$ now follows from Theorem 4.2. In fact it also holds true for $d = d^{*}_{3}$. In this case, the stability for $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ can be concluded by some comparison argument. In addition, there is no positive equilibrium and the equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable in both cases (Ⅰ) and (Ⅱ), by Theorem 2.1 and Theorem 3.3. Hence the trichotomy in Theorem A.2 implies the global convergence to $(0, 0, \bar{v}_{1}, \bar{v}_{2})$. Similarly, we see that the equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is globally asymptotically stable for $d = d^{*}_{1}$ or $d^{*}_{2}$.

5.   Numerical illustrations

We arrange two examples to illustrate the global dynamics of system (1.3), and the bifurcation with respect to the dispersal rate $d$, which are concluded in Theorem 4.2. We also present two more examples to demonstrate that the established scenarios still hold without satisfying condition (P).

Example 1. Consider system (1.3) with $\alpha_1 = 1$, $\alpha_2 = 3$, $\beta_1 = 1.5$ and $\beta_2 = 2.8$, i.e., $\sigma_1 = 0.5$ and $\sigma_2 = 0.2$. Let us examine the conditions in Theorem 4.2(Ⅰ): condition (C): $\alpha_1 = 1 < \beta_1 = 1.5 < \beta_2 = 2.8 < \alpha_2 = 3$ with $\sigma_2 = 0.2 < \sigma_1 = 0.5$; condition $({\rm P})$: $\frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} = 0.8 < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} = 0.948$; $\sigma_2 \beta_2 = 0.56 < \sigma_1 \beta_1 = 0.75$, and $\beta_2-\beta_1 = 1.3\geq \sigma_1+\sigma_2 = 0.7$. We depict in Figure 5 the bifurcation diagram with respect to the dispersal rate $d$. It appears that $d^{*}_{3}\cong 1.22$, which is consistent with Theorems 2.1(ⅰ) and 4.2(Ⅰ): $\frac{\sigma_1 \sigma_2}{\sigma_1- \sigma_2} = 0.333 < d^{*}_{3} < \frac{\sigma_1}{\sigma_1-\sigma_2}(\alpha_2-\alpha_1) = 3.33$. The globally stable positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ exists for $d < d^{*}_{3}$ and collides with the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ at $d = d^{*}_{3}$. For $d\geq d^{*}_{3}$, the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ becomes globally attractive.

Example 2. Consider system (1.3) with $\alpha_1 = 1$, $\alpha_2 = 3$, $\beta_1 = 1.7$ and $\beta_2 = 2.5$, i.e., $\sigma_1 = 0.7$ and $\sigma_2 = 0.5$. Let us examine the conditions in Theorem 4.2(Ⅱ): condition (C): $\alpha_1 = 1 < \beta_1 = 1.7 < \beta_2 = 2.5 < \alpha_2 = 3$ with $\sigma_2 = 0.5 < \sigma_1 = 0.7$; condition $({\rm P})$: $\frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} = 1.042 < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} = 1.35$; $\sigma_2 \beta_2 = 1.25 > \sigma_1 \beta_1 = 1.19$. The bifurcation diagram with respect to the dispersal rate $d$ is depicted in Figure 6. It appears that $d^{*}_{1}\cong 0.91$, $d^{*}_{2} \cong 1.92$, $d^{*}_{3}\cong 4.15$, which are consistent with Theorem 2.1(ⅱ) and 4.2(Ⅱ): $\frac{\sigma_1 \sigma_2\alpha_{1}\alpha_{2}}{\sigma_2\alpha_{2}^2-\sigma_1\alpha_{1}^2 } = 0.276 < d^{*}_{1} < \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} = 1.35$, $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} = 1.75 < d^{*}_{2} < \frac{\sigma_{1}\sqrt{\sigma_{1}\sigma_{2}}+\sigma_{1}\sigma_{2}}{\sigma_{1}^{2}-\sigma_{2}^{2}}(\alpha_{2}-\alpha_{1}) = 6.36$, and $\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}-\sigma_{2}} = 1.75 < d^{*}_{3} < \frac{\sigma_{1}}{\sigma_{1}-\sigma_{2}}(\alpha_{2}-\alpha_{1}) = 7$. The globally stable positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ exists for $d < d^{*}_{1}$ and collides with the semitrivial equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ at $d = d^{*}_{1}$. For $d^{*}_{1}\leq d \leq d^{*}_{2}$, the equilibrium $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ becomes globally attractive and for $d^{*}_{2} < d < d^{*}_{3}$, the globally stable positive equilibrium $(u^{*}_{1}, u^{*}_{2}, v^{*}_{1}, v^{*}_{2})$ exists. For $d \geq d^{*}_{3}$, the semitrivial equilibrium $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ becomes globally attractive.

Example 3. Consider system (1.3) with $\alpha_1 = 1$, $\alpha_2 = 3$, $\beta_1 = 1.4$ and $\beta_2 = 2.85$, i.e., $\sigma_1 = 0.4$ and $\sigma_2 = 0.15$. For such parameter values, condition (C) holds: $\alpha_1 = 1 < \beta_1 = 1.4 < \beta_2 = 2.85 < \alpha_2 = 3$ with $\sigma_2 = 0.15 < \sigma_1 = 0.4$. In addition, $\sigma_2 \beta_2 = 0.4275 < \sigma_1 \beta_1 = 0.56$. Such parameter values violate condition $({\rm P})$, as $\frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} = 0.777 > \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} = 0.551$. Nevertheless, the same dynamical scenario as Example 1 takes place, as seen in Figure 7.

Example 4. Consider system (1.3) with $\alpha_1 = 1$, $\alpha_2 = 3$, $\beta_1 = 1.35$ and $\beta_2 = 2.8$, i.e., $\sigma_1 = 0.35$ and $\sigma_2 = 0.2$. With such parameter values, condition (C) holds: $\alpha_1 = 1 < \beta_1 = 1.35 < \beta_2 = 2.8 < \alpha_2 = 3$ with $\sigma_2 = 0.2 < \sigma_1 = 0.35$; $\sigma_2 \beta_2 = 0.56 > \sigma_1 \beta_1 = 0.4725$. These parameter values violate condition $({\rm P})$, as $\frac{\sigma_{2}\beta_{2}}{\sigma_{1}+\sigma_{2}} = 1.0182 > \frac{\sigma_1\sigma_2\beta_{1}\beta_{2}}{\sigma_2\beta_{2}^2-\sigma_1\beta_{1}^2} = 0.2845$. Nevertheless, the dynamical scenario shown in Figure 8 remains identical to Example 2.

6.   Discussions and conclusions

We have exhibited the global dynamics for a model on two-species competition in a two-patch environment, under certain conditions. The main condition (C): $\alpha_{1} < \beta_{1} < \beta_{2} < \alpha_{2}$, $(\beta_{1}+ \beta_{2})-(\alpha_{1}+\alpha_{2}) = \sigma_{1}-\sigma_{2} > 0$, indicates that the birth rate of $u$-species in the second patch is the largest among all birth rates of two species on two patches, yet the average birth rate of $v$-species is larger than $u$-species. This means that the birth rate for $v$-species is larger than $u$-species in the first patch. The present investigation exploited analytically two dynamical scenarios for such competition, as demonstrated in Examples 1 and 2, respectively. The first scenario takes place under $\sigma_{2}\beta_{2} < \sigma_{1}\beta_{1}$. As expressed by $\frac{\sigma_{1}}{\sigma_2} > \frac{\beta_{2}}{\beta_{1}} > 1$, it indicates that the value of $\sigma_1$ is larger than the value of $\sigma_2$ in a way that its ratio exceeds the ratio of $\beta_2$ over $\beta_1$. This includes the situation that $\sigma_1$ is much bigger than $\sigma_2$, which is denoted by $(\beta_{1}+ \beta_{2})\gg (\alpha_{1}+\alpha_{2})$. The second scenario comes about under $\sigma_{2}\beta_{2} > \sigma_{1}\beta_{1}$. On the contrary, as expressed by $1 < \frac{\sigma_{1}}{\sigma_2} < \frac{\beta_{2}}{\beta_{1}}$, it indicates that the value of $\sigma_1$ may be merely a little over the value of $\sigma_2$, depending on the ratio of $\beta_2$ over $\beta_1$. In this case, the average birth rate of $v$-species may be merely a little more than and close to the average birth rate of $u$-species; we denote this situation by $(\beta_{1}+ \beta_{2})\approx (\alpha_{1}+\alpha_{2})$.

In the first case, including the sense $(\beta_{1}+ \beta_{2})\gg (\alpha_{1}+\alpha_{2})$, coexistence of two species occurs for dispersal rate $d < d^{*}_{3}$, and $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ is globally attractive for $d \geq d^{*}_{3}$, where $d^{*}_{3}$ has been estimated by system parameters. In this situation, $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is unstable for any $d > 0$ and an eigenvalue of the linearized system at $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ changes from positive to negative as $d$, being increasing from $0$, exceeds $d^{*}_{3}$, and $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ becomes stable for $d \geq d^{*}_{3}$.

In the second case, including the sense $(\beta_{1}+ \beta_{2})\approx (\alpha_{1}+\alpha_{2})$, the coexistence of two species takes place for $d < d^{*}_{1}$ or $d^{*}_{2} < d < d^{*}_{3}$, $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ is globally attractive for $d^{*}_{1}\leq d \leq d^{*}_{2}$, and $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ becomes globally attractive for $d \geq d^{*}_{3}$, where $d^{*}_{1}, d^{*}_{2}, d^{*}_{3}$ have been estimated. An eigenvalue of the linearized system at $(\bar{u}_{1}, \bar{u}_{2}, 0, 0)$ changes from positive to negative at $d^{*}_{1}$, and then back to positive at $d^{*}_{2}$. In addition, an eigenvalue of the linearized system at $(0, 0, \bar{v}_{1}, \bar{v}_{2})$ changes from positive to negative at $d^{*}_{3}$, and $d^{*}_{2} < d^{*}_{3}$.

Our analytical investigation on this model strongly suggests that, in high-dispersal situations, one species will prevail if its average birth rate is larger than the other competing species, whereas in low-dispersal situations, the two species can coexist or one species that has the greatest birth rate in one patch among all species and patches will be able to persist and drive the other species to extinction, even though its average birth rate is lower. Such findings may illuminate some insights into how species learn to compete and point out the evolution directions.

Condition (C) is a basic assumption for the present results. Although there are additional conditions $({\rm P})$ and $\beta_2 - \beta_1 \geq \sigma_1+\sigma_2$, due to mathematical technicality, it is believed that such scenarios remain true under condition (C) only. However, it is very difficult to remove these additional conditions, as the algebraic operations involving five parameters are rather involved. In Examples 3, 4, we have demonstrated exactly the same dynamical scenarios for parameter values which do not satisfy condition $({\rm P})$.

To compare our results with those in ^[19], we set $\sigma_1 = \xi\sigma_2$, $\xi > 1$, according to condition (C). The resource difference between two species can be depicted as $(\sigma_1, -\sigma_2)$ among two patches, where $\beta_1-\alpha_1 = \sigma_1 > 0$ means that $v$-species has an advantage over $v$-species in competing the resource in patch-1, while $\beta_2-\alpha_2 = -\sigma_2 < 0$ means that it is disadvantageous for $v$-species to compete with $u$-species for the resource in patch-2. We rewrite it as $\sigma_2(\xi, -1)$ with fixed $\xi > 1$, and now the value of $\sigma_2$ measures the difference between two species and resembles the value of $\tau$ in ^[19]. We accordingly rewrite the conditions in Theorem 4.2 to explore how the magnitude of resource difference affects the invasion of mutant species:

Note that the criteria in Theorem 4.2(Ⅱ) imply $\beta_2-\beta_1 < \sigma_1+\sigma_2$. Therefore, by increasing the dispersal rate $d$, the global convergence of system (1.1) switches in case (Ⅰ) of Theorem 4.2 from the coexistence to extinction of $u$-species when $\sigma_2^{*} < \sigma_2\leq\sigma_2^{**}$ and $\xi > \frac{\beta_2}{\beta_1}$; on the other hand, in case (Ⅱ), the dynamics switches three times from global convergence to the coexistence to extinction of (mutant) $v$-species, again to the coexistence and then the persistence of $v$-species, when $\sigma_2 > \sigma_2^{**}$ and $\xi < \frac{\beta_2}{\beta_1}$. This result enhances the understanding on the dynamics of competitive species from the viewpoint of patchy habitat in the following aspects: Compared to concluding global convergence under small magnitude of spatial heterogeneity ($\tau$) in [19,Theorem 1.2], our result in Theorem 4.2 admits a large range of magnitudes ($\sigma_2$) depicting spatial heterogeneity. The multiple stability switches in Theorem 4.2 are global dynamics, as compared to local dynamics in [19,Theorem 1.1].

Acknowledgements

The authors are grateful to Chao-Nien Chen for suggesting to study the case with average birth rates of opposite order, and to Yuan Lou for very helpful discussions. The authors are supported, in part, by the Ministry of Science and Technology, Taiwan.

Conflict of interest

All authors declare no conflicts of interest in this paper.

Appendix Ⅰ.   Monotone dynamics theory

For reader's convenience, we review some theory on monotone dynamical systems from ^[17] and ^[28]. Denote by $\mathbb{R}^{n}_{+} = \{\mathbf{x} = (x_1, \ldots, x_n)\in \mathbb{R}^{n}: x_{i}\geq 0, 1\leq i\leq n\}$ the first octant of $\mathbb{R}^{n}$. For $\mathbf{x}, \mathbf{y}\in \mathbb{R}^{n}_{+}$, define the following order: $\mathbf{x} \leq_{m} \mathbf{y}$ if $\mathbf{y}-\mathbf{x}\in K_m$, and $\mathbf{x} \ll_m \mathbf{y}$ whenever $\mathbf{y}-\mathbf{x}\in {\rm Int}K_m$, where

If $\mathbf{x} \leq_m \mathbf{y}$, we define $[\mathbf{x}, \mathbf{y}]_m = \{\mathbf{z}\in \mathbb{R}^{n}_{+}: \mathbf{x} \leq_m \mathbf{z} \leq_m \mathbf{y} \}$ and $(\mathbf{x}, \mathbf{y})_m = \{\mathbf{z}\in \mathbb{R}^{n}_{+}: \mathbf{x} \ll_m \mathbf{z} \ll_m \mathbf{y} \}$.

A semiflow $\phi$ is said to be of type-$K$ monotone with respect to $K_m$, provided

A system of ODEs $\dot{\mathbf{x}} = {\bf f}(\mathbf{x})$ is called a type-$K$ monotone system with respect to $K_m$ if the Jacobian matrix of ${\bf f}$ takes the form

where $A_{1}$ is an $k\times k$ matrix, $A_{4}$ an $(n-k)\times (n-k)$ matrix, $A_{2}$ an $k\times (n-k)$ matrix, $A_{3}$ an $(n-k)\times k$ matrix, every off-diagonal entry of $A_{1}$ and $A_{4}$ is nonnegative, and $A_{2}$ and $A_{3}$ are nonnegative matrices, for some $k$ with $1 \leq k\leq n$. It was shown in ^[27] that the flow $\phi_{t}(\mathbf{x})$ generated by the type-$K$ monotone system is type-$K$ monotone with respect to the cone $K_m$, i.e., if $\mathbf{x}, \mathbf{y}\in \mathbb{R}^{n}_{+}$ with $x_{i} \leq y_{i}$ for $1 \leq i \leq k$ and $x_{j} \geq y_{j}$ for $k+1 \leq j \leq n$, then for any $t > 0$, $(\phi_{t}(\mathbf{x}))_{i} \leq (\phi_{t}(\mathbf{y}))_{i}$ for $1 \leq i \leq k$ and $(\phi_{t}(\mathbf{x}))_{j} \geq (\phi_{t}(\mathbf{y}))_{j}$ for $k+1 \leq j \leq n$.

System (1.3) is a type-$K$ monotone system with respect to

since its Jacobian matrix is

For system (1.3), let us denote by $e_{\mathbf{0}}: = (0, 0, 0, 0)$ the trivial equilibrium, by $e_{\bar{\mathbf{u}}}: = (\bar{u}_{1}, \bar{u}_{2}, 0, 0)$, and $e_{\bar{\mathbf{v}}}: = (0, 0, \bar{v}_{1}, \bar{v}_{2})$, $\bar{u}_{i}$, $\bar{v}_{i} > 0$, $i = 1, 2$, the semitrivial equilibria. If $(u_{1}, u_{2}, v_{1}, v_{2}) \in \mathbb{R}^{4}_{+}$, then $(0, 0, v_{1}, v_{2})\leq_{m} (u_{1}, u_{2}, v_{1}, v_{2}) \leq_{m} (u_{1}, u_{2}, 0, 0)$, and therefore,

for all $t \geq 0$. Since $\phi_{t}((0, 0, v_{1}, v_{2}))\rightarrow e_{\bar{\mathbf{v}}}$ and $\phi_{t}((u_{1}, u_{2}, 0, 0))\rightarrow e_{\bar{\mathbf{u}}}$ as $t \rightarrow \infty$, for $(u_{1}, u_{2}, v_{1}, v_{2}) \in \mathbb{R}^{4}_{+}$, and $u_{1}+ u_{2} > 0$, $v_{1}+ v_{2} > 0$, it follows that all points in $\mathbb{R}^{4}_{+}$ are attracted to the set

If $\mathbf{w} = (u_{1}, u_{2}, v_{1}, v_{2})$ with $u_{1}, u_{2}, v_{1}, v_{2} > 0$, then $\phi_{t}(\mathbf{w})\gg 0$ for $t > 0$. Define $E$ and $E^{+}$ the sets of all nonnegative equilibria and all positive equilibria for $\phi_{t}$, respectively. Obviously, $[e_{\bar{\mathbf{v}}}, e_{\bar{\mathbf{u}}}]_{m}$ contains $E$ and $e_{*}\in (e_{\bar{\mathbf{v}}}, e_{\bar{\mathbf{u}}})_{m}$ for any $e_{*}\in E^{+}$. The following theorem restates Corollary 4.4.3 in ^[28] for system (1.3), see also ^[27,31].

Theorem A.1. If $e_{\bar{\mathbf{u}}}$ and $e_{\bar{\mathbf{v}}}$ are both linearly unstable, then system (1.3) is permanent. More precisely, there exist positive equilibria $e_{*}$ and $e_{**}$, not necessarily distinct, satisfying

The order interval

attracts all solutions evolved from $\mathbf{w} = (u_{1}, u_{2}, v_{1}, v_{2}) \in {\mathbb R}^4_+$, with $u_{1}+ u_{2} > 0$ and $v_{1}+ v_{2} > 0$. In particular, if $e_{**} = e_{*}$, then $e_{*}$ attracts all such solutions.

It was shown in ^[17] that, for models of two competing species, either there is a positive equilibrium representing coexistence of two species, or one species drives the other to extinction. Note that system (1.3) satisfies conditions (H1)-(H4) in ^[17], and thus Theorem B in ^[17] can be restated as follows.

Theorem A.2. Consider system (1.3). The $\omega$-limit set of every orbit evolved from a point in $\mathbb{R}^{4}_{+}$ is contained in $\Gamma$ and exactly one of the following holds:

(a) There exists a positive equilibrium $e_{*}$ of in $\Gamma$.

(b) $\phi_{t}(\mathbf{w})\rightarrow e_{\bar{\mathbf{u}}}$ as $t\rightarrow \infty$, for every $\mathbf{w} = (u_{1}, u_{2}, v_{1}, v_{2})\in \Gamma$ with $u_{1}+ u_{2} > 0$.

(c) $\phi_{t}(\mathbf{w})\rightarrow e_{\bar{\mathbf{v}}}$ as $t\rightarrow \infty$, for every $\mathbf{w} = (u_{1}, u_{2}, v_{1}, v_{2})\in \Gamma$ with $v_{1}+ v_{2} > 0$.

In addition, if (b) or (c) holds, then either $\phi_{t}(\mathbf{w})\rightarrow e_{\bar{\mathbf{u}}}$ or $\phi_{t}(\mathbf{w})\rightarrow e_{\bar{\mathbf{v}}}$, as $t\rightarrow \infty$, for $\mathbf{w} = (u_{1}, u_{2}, v_{1}, v_{2})\in \mathbb{R}^{4}_{+}\setminus \Gamma$.

Appendix Ⅱ.    

We recall some qualitative properties of the semitrivial equilibria for system (1.3) in ^[24]. The following results are independent of the order between $\sigma_1$ and $\sigma_2$.

Proposition A.3 (Proposition 3.7 ^[24]). If $\alpha_1 < \alpha_2$, the following hold for all $d > 0$.

(ⅰ) $\alpha_{1} < \bar{u}_{1} < \bar{u}_{2} < \alpha_{2}$.

(ⅱ) $(\alpha_{1}-\bar{u}_{1})- (\alpha_{2}-\bar{u}_{2}) = \frac{d(\bar{u}_1^2-\bar{u}_2^2)}{\bar{u}_1\bar{u}_2} < 0$, $(\alpha_{1}-\bar{u}_{1})+ (\alpha_{2}-\bar{u}_{2}) = d\left[2-\left(\frac{\bar{u}_2}{\bar{u}_1}+\frac{\bar{u}_1}{\bar{u}_2}\right)\right] < 0$.

(ⅲ) $\alpha_{1} < \bar{u}_{1} < \frac{\alpha_{1}+\alpha_{2}}{2} < \bar{u}_{2} < \alpha_{2}$.

Proposition A.4 (Proposition 3.8 ^[24]). If $\beta_1 < \beta_2$, the following hold for all $d > 0$.

(ⅰ) $\beta_{1} < \bar{v}_{1} < \bar{v}_{2} < \beta_{2}$.

(ⅱ) $(\beta_{1}-\bar{v}_{1})- (\beta_{2}-\bar{v}_{2}) = \frac{d(\bar{v}_1^2-\bar{v}_2^2)}{\bar{v}_1\bar{v}_2} < 0$, $(\beta_{1}-\bar{v}_{1})+ (\beta_{2}-\bar{v}_{2}) = d\left[2-\left(\frac{\bar{v}_2}{\bar{v}_1}+\frac{\bar{v}_1}{\bar{v}_2}\right)\right] < 0$.

(ⅲ) $\beta_{1} < \bar{v}_{1} < \frac{\beta_{1}+\beta_{2}}{2} < \bar{v}_{2} < \beta_{2}$.

Proposition A.5 (Proposition 3.9 ^[24]). If $\alpha_1 < \alpha_2$, the following hold:

(ⅰ) $\bar{u}_{1}, \bar{u}_{2}\rightarrow \frac{\alpha_{1}+\alpha_{2}}{2}$ as $d\rightarrow\infty$.

(ⅱ) $d$ is strictly decreasing with respect to $\bar{u}_{2}$ on $(\frac{\alpha_{1}+\alpha_{2}}{2}, \alpha_{2})$, and $d$ is strictly increasing with respect to $\bar{u}_{1}$ on $(\alpha_{1}, \frac{\alpha_{1}+\alpha_{2}}{2})$.

Proposition A.6 (Proposition 3.10 ^[24]). If $\beta_1 < \beta_2$, the following hold:

(ⅰ) $\bar{v}_{1}, \bar{v}_{2}\rightarrow \frac{\beta_{1}+\beta_{2}}{2}$ as $d\rightarrow\infty$.

(ⅱ) $d$ is strictly decreasing with respect to $\bar{v}_{2}$ on $(\frac{\beta_{1}+\beta_{2}}{2}, \beta_{2})$, and $d$ is strictly increasing with respect to $\bar{v}_{1}$ on $(\beta_{1}, \frac{\beta_{1}+\beta_{2}}{2})$.