A heterogeneous parasitic-mutualistic model of mistletoes and birds on a periodically evolving domain

1.   Introduction and model formulation

As a unique vector-borne parasite, mistletoe is not only parasitic and commensal, but also mutulistic with its vectors, that is, the avian seed-dispersers. In fact, Mistletoes always infect vascular plants ranging from cacti to pines, and prevail in many areas ^[1]. Although mistletoes are always thought as pests that kill trees and depreciate natural habitats, more and more recent investigators have gradually recognized them as an indispensable ecological keystone species, which contribute greatly to biodiversity by providing high quality food and habitat for a broad range of animals living in woodlands and forests worldwide ^[2].

Due to so special nature of the interaction between mistletoes and their bird vectors, the model of mistletoes and birds has received impressive attention from researchers in many disciplines. In particular, in the earlier study of Liu et al. ^[3], they first proposed a single species model to investigate the dynamics of mistletoes, in which the population of the birds was assumed to be constant, and lived in a common isolated and fixed habitat with mistletoes. Different from ^[3], recently, Wang et al. ^[4] built a more complex model to incorporate the spatial dispersal, the maturation delay of mistletoes and homogeneous spatial-temporal interaction between mistletoes and birds in a fixed spatial domain. By the different choices of the dispersal fashion of mistletoe fruits by birds, Wang et al. ^[4] obtained some detailed information about the spatial patterns of mistletoes and birds, which are extremely beneficial for people to understand the long time dynamics of mistletoes and birds world. Following this pioneering work, Wang et al. ^[5] still explored the spatial spreading behaviors of the same model with some simpler assumptions, and derived the existences of asymptotic spreading speed and traveling waves by monotone semiflow theory.

Although all aforementioned studies are developed on time-independent domains with homogeneous hypothesis on the spatial-temporal environment, one must note that in reality, either the habit that the creature living in or the environment they undergo always keeps changing as time evolving. For example, many birds periodically migrate between different latitudes, where both the food resource and the mortality vary obviously. In fact, there were more and more literatures concerning the asymptotic behaviors of the solutions to the reaction-diffusion systems on some changing domains, and we refer to ^[6,7] for the pattern formation on a growing domain, ^[8,9] for the species model on finite or infinite growing domains, respectively, ^[10] for the logistic equation on a periodically evolving domain, and ^[11,12] for the epidemic systems on a periodically evolving domain.

Motivated by aforementioned studies and the periodicity of birds' territory, we will investigate a heterogeneous parasitic-mutualistic model of mistletoes and birds in a periodically evolving domain, and reveal some new phenomena caused by the domain evolution in the model.

For such aims, we start with some fundamental assumptions for our model. In accordance to ^[10], let $\Omega(t)\subset \mathbb{R}^n$ be a bounded and simply connected domain that periodically evolves over time $t\geq0$ with smooth boundary $\partial\Omega(t)$. Denote $B(t, X(t))$ and $M(t, X(t))$, respectively, be the population densities of birds and mistletoes at time $t$ and spatial position $X(t)\in\Omega(t)$. In the light of ^[4,5] and Reynolds transport theorem ^[13], we will focus on the following dimensionless reaction-diffusion equation

with the initial functions defined on the initial domain $\Omega_0: = \Omega(0)$. In the mistletoes and birds world, $\bar{\alpha}(t, X)$ is the rate of successfully attaching the mistletoe seeds to birds, which is equivalent to the rate of parasites transmission, while $\bar{\delta}(t, X)$ is the mortality rate of the mistletoes. Meanwhile, $\bar{\beta}(t, X)$ is the conversion rate, at which energy is transformed from mistletoes fruits birds eaten into birds population, and $\bar{r}(t, X)$ is the abundance of the food resources other than mistletoes fruits. Moreover, $d_M$ and $d_B$ represent the random diffusivity of mistletoes and birds, respectively, which satisfy $0 < d_M\ll d_B$ attributed to the tremendous scale difference between their territories. On the other hand, $\vec{\nu}$ is the flow velocity yielded by the domain evolving, the advection terms $\vec{\nu}\cdot\nabla M$ and $\vec{\nu}\cdot\nabla B$ represent the transport of populations determined by $\vec{\nu}$, and $M(\nabla\cdot\vec{\nu})$ and $B(\nabla\cdot\vec{\nu})$ are the annihilation term due to local volume changes ^[6,7,14].

To circumvent the complexities due to the terms of dilution and advection, we adopt the transformation between Eulerian coordinate and Lagrangian coordinate in fluid mechanics to modify problem (1.1) from the evolving domain into the fixed domain. Therefore, we first suppose that domain evolves periodically and isotropically, that is,

where $x$ and $X$ are the spatial coordinates of the initial domain $\Omega_0$ and evolving domain $\Omega(t)$, respectively; the evolving ratio $\kappa(t)\in C^1([0, T], (0, \infty))$ is a T-periodic function, i.e., $\kappa(t) = \kappa(t+T)$, and satisfies $\kappa(0) = 1$ and $0 < \kappa(t)\leq \kappa^*$ for all $t\geq0$, where $\kappa^*$ is a positive constant. Moreover, without loss of generality, we also assume that the flow velocity is equivalent to the changing velocity of the domain, i.e., $\vec{\nu} = \dot{X}(t)$. As a result, we have

Hence, redefine

then problem (1.2) and (1.3) yield that

Meanwhile, redefining

as well, problem (1.1) is thus transformed into

with the nontrivial, bounded and continuous initial conditions

Here, we still suppose that $\alpha(t, x), \delta(t, x), r(t, x)$, and $\beta(t, x)\in C^{\frac{\theta}{2}, \theta}([0, \infty)\times\overline{\Omega_0})(\theta\in(0, 1))$ are positive and T-periodic with respect to $t$. Meanwhile, suppose that

on $[0, \infty)\times\overline{\Omega_0}$, where $z = \alpha, \delta, r, \beta$; $z_*$ and $z^*$ are constants.

Additionally, by extending the definition to be zero either $m = 0$ or $b = 0$, we can reconstruct the reaction function $\frac{MB}{M+B}$ as a locally Lipschitz continuous one in the entire first quadrant. Hence, in view of all the assumptions above and the uniform estimates in Lemma 3.4, the standard regularity theory ^[15,16] indicates that problem (1.4)–(1.5) have a unique global solution $(m, b)\in [C^{1+\frac{\theta}{2}, 2+\theta}([0, \infty)\times\overline{\Omega_0})]^2$. Hence, the primary objective of the current article is to investigate the persistence, extinction and coexistence of the birds and mistletoes populations in problem (1.4)–(1.5), and to reveal the important influence and new phenomena caused by the domain evolution. The key technique we adopt here is the spatial-temporal risk index that is similar to the basic reproduction number in epidemiology, which will be established by the principal eigenvalue theory of periodic reaction-diffusion equations.

The remainder of our article is arranged in the following way. In Section 2, we first limit ourself to examine the diffusive dynamics in birds world without mistletoes, and present a fundamental extinction-persistence dichotomy distinguished by the associated spatial-temporal risk index $R^B_0$. Section 3 is devoted to comprehensively investigating the dynamics behaviors in the mistletoes-birds world, in which we establish the core spatial-temporal risk index $R^M_0$, then obtain the extinction-coexistence dichotomy distinguished by $R^M_0$, and explore the impact of evolving domain on the viability of mistletoes. Finally, numerical simulations are also exhibited for some specific cases in Section 4 to illustrate our theoretical conclusions.

2.   The diffusive dynamics in birds world

In this section, we first explore the diffusive dynamics in birds world without mistletoes, and present a fundamental extinction-persistence dichotomy distinguished by the spatial-temporal risk index $R^B_0$, which plays a role being similar to the basic reproduction number in epidemiology.

In fact, provided $m\equiv0$, problem (1.4)–(1.5) become

To further analyze the asymptotic behavior of the solution of problem (2.1), we focus on the periodic parabolic eigenvalue problem

Denote $R^B_0(\kappa): = \mu_0$, where $\mu_0$ is the principal eigenvalue of problem (2.2). We have the following key results.

Lemma 2.1. $sign(1-R^B_0(\kappa)) = sign\lambda_0$, where $\lambda_0$ is the principal eigenvalue of the following eigenvalue problem

Proof. For any fixed $\mu > 0$, consider the eigenvalue problem

and denote its principal eigenvalue as $\Lambda_0(\mu)$. It is well-known ^[17] that $\Lambda_0(\mu)$ is continuous and strictly increasing with respect to $\mu$. Moreover, the uniqueness of principal eigenvalue implies that $\lambda_0 = \Lambda_0(1)$ and $\Lambda_0(\mu_0) = 0$.

On the other hand, since $\Lambda_0(\mu)$ satisfies $\lim_{\mu\rightarrow0^+}\Lambda_0(\mu) < 0$ and $\lim_{\mu\rightarrow\infty}\Lambda_0(\mu) > 0$ ^[17], one can deduce from the monotonicity of $\Lambda_0(\mu)$ that $R^B_0(\kappa) = \mu_0$ is the unique positive solution for the equation $\Lambda_0(\mu) = 0$. Due to

the monotonicity also yields that $sign(1-R^B_0(\kappa)) = sign\lambda_0$, and the proof is completed.

Therefore, completely identical to ^[10], we obtain the following existence and attractivity results of the periodic solutions of problem (2.1) in the evolving domain.

Theorem 2.2. Let $b(t, x;b_0)$ be the solution of problem (2.1) with nonnegative, bounded and continuous initial conditions $b_0(x)\geq0$ on $\overline{\Omega_0}$.

(i) If $R^B_0(\kappa)\leq1$, then the trivial solution $b(t, x)\equiv0$ is globally asymptotically stable in the sense that

holds uniformly on $[0, T]\times\overline{\Omega_0}$;

(ii) If $R^B_0(\kappa) > 1$, then problem (2.1) has a positive and T-periodic solution $b^*(t, x)$, which is globally asymptotically stable for all nontrivial $b_0$ in the sense that

holds uniformly on $[0, T]\times\overline{\Omega_0}$.

3.   The diffusive dynamics in the mistletoes-birds world

Since the case that $R^B_0(\kappa)\leq1$ could lead to some trivial extinction for small invasion populations, throughout this section, we focus on discussing the diffusive dynamics of mistletoes under the influence of birds for the case $R^B_0(\kappa) > 1$.

3.1.   The spatial-temporal risk index $R^M_{0}$

We will introduce the spatial-temporal risk index $R^M_0(\kappa)$ associated with the problem (1.4)–(1.5), which is a similar role as index $R^B_0(\kappa)$, while it will be constructed by a more epidemic fashion. The readers are also referred to the similar conception defined by Lin et al. ^[18,19] for the free boundary problems.

For such aim, we first consider the linearized equation for problem (1.4)–(1.5) at point $(m, b) = (0, b^*)$ as follows

Let $d(t) = \frac{d_M}{\kappa^2(t)}$ and $\eta(t, x) = \delta(t, x)+n\frac{\dot{\kappa}(t)}{\kappa(t)}$, then the first equation of problem (3.1) yields

We first consider the case that $\eta(t, x)\geq0$ for any $t\geq0$ and $x\in\overline{\Omega_0}$.

Let $\mathcal{C}_T$ be all the continuous and T-periodic functions from $\mathbb{R}$ to $C(\overline{\Omega_0}, \mathbb{R})$, which is equipped with the positive cone $\mathcal{C}^+_T: = \{\zeta\in \mathcal{C}_T:\zeta(t)(x)\geq0, \forall t\in \mathbb{R}, x\in\overline{\Omega_0}\}$ and maximum norm $||\cdot||$. It is easy to see that $\mathcal{C}_T$ is an ordered Banach space. We denote $\zeta(t, x): = \zeta(t)(x)$ for any $\zeta\in \mathcal{C}_T$.

Besides, let $A(t, \tau)$ be the evolution operator of problem

Due to the standard semigroup theory, we know that there exist positive constants $K$ and $c_0$ such that

Now, assume that $\zeta\in \mathcal{C}_T$, and let $\zeta(\tau, x)$ be the density distribution of mistletoes individuals at the time $\tau$ and spatial location $x\in \overline{\Omega_0}$. Then the term $\alpha(\tau, x)\zeta(t, x)$ is just the density distribution of the new individuals engendered by the old individuals introduced at time $\tau$. Thus, for any $t\geq \tau$, we let $A(t, \tau)\alpha(\tau, x)\zeta(\tau, x)$ be the density distribution at location $x$ of the individuals, who are newly introduced at time $\tau$ and still are survival at time $t$. Therefore, the following expression

is called the accumulative density distribution of the new mistletoes individuals at time $t$ and location $x$, and they are engendered by those individuals $\zeta(\tau, x)$ introduced at all the time before $t$.

According to the statements in ^[20] and ^[21], we are going to define the operator $\mathfrak{L}: \mathcal{C}_T\rightarrow \mathcal{C}_T$ as the next generation operator, where

It is clear that $\mathfrak{L}$ is continuous, positive and compact on $\mathcal{C}_T$. Hence, similar as in ^[21], we choose the spectral radius $r(\mathfrak{L})$ to be defined as the basic reproduction number to problem (1.4)–(1.5), i.e., $R_0: = r(\mathfrak{L})$.

Furthermore, we have the following significant conclusions.

Lemma 3.1. (i) $R^M_0 = \mu^M_0$ and where $\mu^M_0$ is the principal eigenvalue of the following periodic parabolic eigenvalue problem

(ii) $sign(1-R^M_0) = sign\lambda^M_0$, where $\lambda^M_0$ is the principal eigenvalue of the following eigenvalue problem

On the other hand, for the case that $\eta(t, x) < 0$ for any $t\geq0$ and $x\in\overline{\Omega_0}$, we can still adopt the completely identical method used by Lemma 2.1 in Section 2 to obtain Lemma 3.1, and we omit the details for simplicity.

Remark 3.2. To better explore the association between $\kappa(t)$ and $R^M_0$, we denote $R^M_0 = R^M_0(\kappa)$ and let $\overline{\kappa^{-2}} = \frac{1}{T}\int^T_0\frac{1}{\kappa^2(t)}\mathrm{d}t$ hereafter.

By Lemma 3.1, we have the following results.

Theorem 3.3. (i) If $\alpha(t, x)\equiv\alpha(t)$, and $\delta(t, x)\equiv\delta(t)$ for all $t\in[0, T]$, then

(ii) If $\alpha(t, x)\equiv\alpha > 0$ and $\delta(t, x)\equiv\delta > 0$ are both constants, then

where $\lambda^M_0$ is the principal eigenvalue of the following problem

Proof. (i) Let $\psi_0(t, x) = p(t)\varphi(x) > 0$ for $(t, x)\in (0, \infty)\times\Omega_0$ be the eigenfunction associating with $R^M_0(\kappa)$. Consequently, problem (3.4) sees

which deduces to

Integrating (3.6) from 0 to $t$ gives

It follows from $\psi(t+T, x) = \psi(t, x)$ that one has $p(T) = p(0)$. Thus, we have

(ii) Due to $\overline{\kappa^{-2}} = \frac{1}{T}\int^T_0\frac{1}{\kappa^2(t)}\mathrm{d}t$, it is clear that we have

The proof is completed.

3.2.   The extinction and coexistence dynamics of mistletoes-birds populations

We first make some preliminaries for the proof of the existence of periodic solution.

Lemma 3.4. There exist positive constants $C_m$ and $C_b$ such that the solution $(m, b)(t, x)$ of problem (1.4)–(1.5) satisfying

holds uniformly on $[0, \infty)\times\overline{\Omega_0}$, provided that $(0, 0)\not\equiv, \leq(m_0, b_0)(x)\leq(C_m, C_b)$ on $\overline{\Omega_0}$.

Proof. First, the maximum principle confirms the strict positivity, and we will prove the right side of (3.7).

In fact, by the second equation of (1.4), we deduce that

Consequently, we obtain

uniformly on $[0, \infty)\times\overline{\Omega_0}$, where $c_\kappa: = \min_{t\in[0, T]}\left\{n\frac{\dot{\kappa}(t)}{\kappa(t)}\right\} < 0$. Denote $C_b: = r^*+\beta^*-c_\kappa$. Then we obtain

uniformly on $[0, \infty)\times\overline{\Omega_0}$ if $||b_0||_\infty\leq C_b$.

Similarly, after assuming that $\delta_*+c_\kappa > 0$, we still have

uniformly on $[0, \infty)\times\overline{\Omega_0}$ if $||m_0||_\infty\leq C_m$ where $C_m: = \frac{\alpha^*C_b}{\delta_*+c_\kappa}$

In brief, we have

uniformly on $[0, \infty)\times\overline{\Omega_0}$, provided that $(0, 0)\not\equiv, \leq(m_0, b_0)(x)\leq(C_m, C_b)$ on $\overline{\Omega_0}$. The proof is finished.

Theorem 3.5. Let $(m, b)(t, x;m_0, b_0)$ be the solution of problem (1.4)–(1.5) with nonnegative, bounded and continuous initial conditions $m_0(x)\geq0$ and $b_0(x)\geq, \not\equiv0$ on $\overline{\Omega_0}$. If $R^M_0(\kappa)\leq1$, then the solution $(0, b^*)$ is globally asymptotically stable in the sense that

holds uniformly on $[0, T]\times\overline{\Omega_0}$.

Proof. For $(t, x)\in[0, T)\times\overline{\Omega_0}$, selecting $m^u(t, x) = Ce^{-\sigma t}\psi(t, x)$, where $\psi(t, x)$ satisfying $\|\psi\|_\infty = 1$ is the normalized eigenfunction associated with $R^M_0(\kappa)$, $0\leq\sigma\leq\alpha(t, x)(\frac{1}{R^M_0}-1)$, and $C > 0$ is a sufficiently large constant. It follows that

Then, $m^u$ is the upper solution of the following problem

Hence, the uniform limit $\lim_{t\rightarrow\infty}m(t, x) = 0$ is derived from the fact that $\lim_{t\rightarrow\infty}m^u(t, x) = 0$ uniformly for $x\in\overline{\Omega_0}$. Furthermore, by the almost parallel method adopted in ^[22], we also can show that $\lim_{i\rightarrow\infty}b(t+iT, x) = b^*(t, x)$ holds uniformly on $[0, T]\times\overline{\Omega_0}$. The proof is completed.

Next, we combine the related eigenvalue problem and ordered upper and lower solutions to explore the periodic steady state coexistence solutions of problem (1.4)–(1.5) and their attractivity. Concretely speaking, we will focus on the following problem

The following definition of upper and lower solutions is fundamental.

Definition 3.6. $(\widetilde{m}, \widetilde{b})(t, x)$ and $(\widehat{m}, \widehat{b})(t, x)$ are called upper and lower solutions to problem (3.8) if

Moreover, $(\widetilde{m}, \widetilde{b})(t, x)$ and $(\widehat{m}, \widehat{b})(t, x)$ are called a pair of ordered upper and lower solutions if they satisfy $(0, 0)\leq(\widehat{m}, \widehat{b})\leq(\widetilde{m}, \widetilde{b})\leq(C_m, C_b)$.

The parallel definition to problem (1.4)–(1.5) could be obtained by similar fashion as well.

For further analysis, we also set

and

where $F$ and $G$ are clearly nondecreasing about $m$ and $b$. Then problem (3.8) can be rewritten as

We now select $(\overline{m}^{(0)}, \overline{b}^{(0)}) = (\widetilde{m}, \widetilde{b})$ and $(\underline{m}^{(0)}, \underline{b}^{(0)}) = (\widehat{m}, \widehat{b})$ as initial functions and construct sequences $\{(\overline{m}^{(i)}, \overline{b}^{(i)})\}$ and $\{(\underline{m}^{(i)}, \underline{b}^{(i)})\}$ $(i = 1, 2, ..., )$ by the following iteration procedure

We first derive the following fundamental properties of the two sequences above.

Lemma 3.7. Assume that ${(\overline{m}^{(i)}, \overline{b}^{(i)})}$ and ${(\underline{m}^{(i')}, \underline{b}^{(i')})}$ are ordered upper and lower solutions of problem (3.8) for any $i$ and $i'$. If there exist

in $\Omega_0$, then ${(\overline{m}^{(i)}, \overline{b}^{(i)})}$ and ${(\underline{m}^{(i')}, \underline{b}^{(i')})}$ are also ordered upper and lower solutions of problem (1.4)–(1.5).

If we still denote the two sequences generated by (3.11) as $\{(\overline{m}^{(i)}, \overline{b}^{(i)})\}$ and $\{(\underline{m}^{(i)}, \underline{b}^{(i)})\}$ with

and $(\widehat{m}, \widehat{b})\leq(m_0(x), b_0(x))\leq(\widetilde{m}, \widetilde{b})$ in $\Omega_0$, then the following result holds.

Lemma 3.8. The two sequences $\{(\overline{m}^{(i)}, \overline{b}^{(i)})\}$ and $\{(\underline{m}^{(i)}, \underline{b}^{(i)})\}$ converge monotonically to a unique solution $(m, b)(t, x)$ of problem (1.4)–(1.5). Moreover,

holds on $[0, \infty)\times\overline{\Omega_0}$.

All the proofs of the lemmas above are routine, so we omit them and refer the details to [23,Theorem B,Lemmas 3.1-3.2].

To present our coexistence results, we need the following key lemma.

Lemma 3.9. The principal eigenvalue $\lambda^M_0$ of problem (3.5) is also an eigenvalue for the following eigenvalue problem with some strict positive eigenfunctions $(\Psi_0, \Phi_0)$

provided that $\lambda^M_0 < 0$.

Proof. Let $(\lambda^M_0, \psi_0)$ be the eigenpair of problem (3.5) with $\lambda^M_0 < 0$ and $\psi_0 > 0$. Then, $(\lambda^M_0, \psi_0)$ satisfies

We then consider the following inhomogeneous problem of $\Phi = \Phi(t, x)$

Since $b^*(t, x)$ solves

the monotonicity of the principal eigenvalue implies that the following problem

has a positive principal eigenvalue $\Lambda_0 > 0$.

Thus, utilizing the positivity of $\beta$ and $\psi_0$ together with [16, Theorem 16.6], we derive that problem (3.15) has a unique solution $\Phi_0(t, x)$ satisfying $\Phi_0(t, x) > 0$ for all $(t, x)\in[0, T]\times\Omega_0$. To sum up, if the principal eigenvalue $\lambda^M_0 < 0$, then it is still an eigenvalue of the eigenvalue problem (3.13) with strict positive eigenfunctions $(\Psi_0, \Phi_0) = (\psi_0, \Phi_0)$. The lemma is proved.

Now, it is the turn to present our main theorem as following.

Theorem 3.10. If $R^M_0(\kappa) > 1$, then the following conclusions hold:

(i) There are a pair of minimal and maximal positive T-periodic solutions $(\underline{m}, \underline{b})\leq(\overline{m}, \overline{b})$ of problem (3.8) over $(0, b^*)$, moreover, if $(\underline{m}, \underline{b})(0, x) = (\overline{m}, \overline{b})(0, x)$, then $(\overline{m}, \overline{b}) = (\underline{m}, \underline{b}): = (m^{\diamond}, b^{\diamond})$ is the unique positive T-periodic solution to problem (3.8);

(ii) Let $(m, b)(t, x;m_0, b_0)$ be the solution of problem (1.4)–(1.5) with bounded and continuous initial conditions $(0, 0)\not\equiv, \leq(m_0, b_0)(x)\leq(C_m, C_b)$ on $\overline{\Omega_0}$, then $(\underline{m}, \underline{b})\leq(\overline{m}, \overline{b})$ is attractive in the sense that

holds uniformly on $[0, T]\times\overline{\Omega_0}$.

Proof. (i) It is clear that $(C_m, C_b)$ and $(0, b^*)$ are ordered upper and lower solutions of problem (3.8). We select the $(\overline{m}^{(0)}, \overline{b}^{(0)}) = (\widetilde{m}, \widetilde{b}) = (C_m, C_b)$ and $(\underline{m}^{(0)}, \underline{b}^{(0)}) = (\widehat{m}, \widehat{b}) = (\gamma\Psi, b^*+\gamma\Phi)$ as initial iteration, where $(\Psi, \Phi)$ is the positive eigenfunction of eigenvalue problem (3.13) associated with $\lambda^M_0 < 0$ given by Lemma 3.9, and $\gamma$ is sufficiently small positive number.

[23, Lemma 3.1] ensures that the sequences $\{(\overline{m}^{(i)}, \overline{b}^{(i)})\}$ and $\{(\underline{m}^{(i)}, \underline{b}^{(i)})\}$ defined by (3.11) have the monotonicity

Therefore, the the monotone convergence theorem [23, Theorem A] implies that there exist a pair of $(\overline{m}, \overline{b})$ and $(\underline{m}, \underline{b})$ satisfying

and

Clearly, $(\underline{m}, \underline{b})$ and $(\overline{m}, \overline{b})$ are T-periodic solutions of problem (3.8) satisfying $(0, b^*)\leq(\underline{m}, \underline{b})\leq(\overline{m}, \overline{b})$.

Next, we claim that $(\underline{m}, \underline{b})$ and $(\overline{m}, \overline{b})$ are minimal and maximal, respectively.

In fact, any T-periodic solution $(m^{**}, b^{**})$ over $(0, b^*)$ satisfies $(\widehat{m}, \widehat{b})\leq(m^{**}, b^{**})\leq(\widetilde{m}, \widetilde{b})$ on $[0, T]\times\overline{\Omega_0}$ if $\gamma > 0$ is sufficiently small. Hence, after choosing $(\widetilde{m}, \widetilde{b})$ and $(m^{**}, b^{**})$ as the initial data for iteration, and operate iteration like problem (3.11), it follows that

that is, $(\overline{m}, \overline{b})$ is the maximal T-periodic solution of problem (3.8). Similarly, we also have

which indicates that $(\underline{m}, \underline{b})$ is the minimal T-periodic solution of problem (3.8) over $(0, b^*)$.

Lastly, the conclusion $(\overline{m}, \overline{b}) = (\underline{m}, \underline{b}) = (m^{\diamond}, b^{\diamond})$ can be derived by the standard existence-uniqueness theorem on the initial-boundary parabolic equations, provided that $(\overline{m}, \overline{b})(0, x) = (\underline{m}, \underline{b})(0, x)$.

(ii) Due to the comparison principle, we first have $b(t, x)\geq \widetilde{b}(t, x)$ on $[0, \infty)\times\overline{\Omega_0}$ for the same initial data, where $\widetilde{b}(t, x)$ is the solution of problem (2.1). Since $R^B_0(\kappa) > 1$, Theorem 2.2 yields that $\lim_{i\rightarrow\infty}\widetilde{b}(t+iT, x) = b^*(t, x)$ holds uniformly on $[0, T]\times\overline{\Omega_0}$. Thus, $\liminf_{i\rightarrow\infty}b(t+iT, x)\geq b^*(t, x)$ holds uniformly on $[0, T]\times\overline{\Omega_0}$. it follows that for any sufficiently small $\varepsilon > 0$, there is a $i^*_\varepsilon > 0$ such that $b(t+iT, x) > b^*(t, x)-\varepsilon$ on $[0, T]\times\overline{\Omega_0}$ for any $i\geq i^*_\varepsilon$.

Set

for any positive integer $i\geq1$. Since

uniformly in $\Omega_0$ for sufficiently small $\varepsilon, \gamma > 0$, Lemma 3.8 implies

on $[0, \infty)\times\overline{\Omega_0}$. In particular,

on $[0, \infty)\times\overline{\Omega_0}$.

Therefore, by choosing $(\widehat{m}, \widehat{b}-\frac{\varepsilon}{2})$ as new initial data $(\underline{m}^{(0)}_\varepsilon, \underline{b}^{(0)}_\varepsilon)$, we still obtain a similar monotone sequences $(\underline{m}^{(i)}_\varepsilon, \underline{b}^{(i)}_\varepsilon)(i = 1, 2, \cdots)$ from (3.11). Moreover, the monotone convergence theorem [23, Theorem A] still yields that $(\underline{m}^{(i)}_\varepsilon, \underline{b}^{(i)}_\varepsilon)$ converge uniformly to a T-periodic solution $(\underline{m}_\varepsilon, \underline{b}_\varepsilon)$ of problem (3.8) as ${i\rightarrow\infty}$, and the later converge uniformly to $(\underline{m}, \underline{b})$ as ${\varepsilon\rightarrow0}$.

By the periodic condition in problem (3.11), we derive

which implies that

in $\Omega_0$. Therefore, we know $(\overline{m}^{(1)}, \overline{b}^{(1)})$ and $(\underline{m}^{(1)}_\varepsilon, \underline{b}^{(1)}_\varepsilon)$ are a pair of order upper and lower solutions to problem (3.8) with initial data $(m_{i^*+1}, b_{i^*+1})(0, x)$ due to Lemma 3.7. By Lemma 3.8 again, the unique solution $(m_{i^*+1}, b_{i^*+1})(t, x)$ of problem (3.8) satisfies

on $[0, \infty)\times\overline{\Omega_0}$. We obtain from the induction principle that for $i\geq1$,

holds on $[0, \infty)\times\overline{\Omega_0}$. Consequently, we have

holds uniformly on $[0, T]\times\overline{\Omega_0}$, and the desired result follows from passing to the limits as ${\varepsilon\rightarrow0}$. The proof is finished.

3.3.   The impact of domain evolution on the viability of mistletoes

The discussion in section 3.2 is fulfilled on the evolving domain, and we will explore the asymptotic behaviors of the related solution on the fixed domain in the sequel. In fact, if $\kappa(t)\equiv1$ and $\Omega(t) = \Omega(0)$, then the underlying domain is constant, and problem (1.4) is transformed to

with the similar nontrivial, nonnegative and continuous initial conditions

The associated periodic steady state coexistence problem is defined by

We still denote the spatia-temporal risk index of problem (3.20)–(3.21) as

by the identical arguments with Lemma 3.1, and we have the following threshold results completely similar to Theorems 3.5 and 3.10.

Theorem 3.11. Let $(u, v)(t, x;m_0, b_0)$ be the solution of problem (3.20)–(3.21) with nonnegative, bounded and continuous initial conditions $u_0(x)\geq0$ and $v_0(x)\geq, \not\equiv0$ on $\overline{\Omega}(0)$, then the following conclusions hold:

(i) If $\Re^M_0\leq1$, then the solution $(0, v^*)$ of problem (3.22) is globally asymptotically stable in the sense that

holds uniformly on $[0, T]\times\overline{\Omega}(0)$, where $v^*(t, x)$ is the unique solution of the periodic parabolic problem

(ii) If $\Re^M_0 > 1$, then problem (3.22) has a pair of maximal and minimal positive positive T-periodic solution $(\underline{u}, \underline{v})\leq(\overline{u}, \overline{v})$ over than $(0, b^\star)$; moreover, if $(\underline{u}, \underline{v})(0, x) = (\overline{u}, \overline{v})(0, x)$, then $(\overline{u}, \overline{v}) = (\underline{u}, \underline{v}): = (u^{\diamond}, v^{\diamond})$ is the unique positive T-periodic solution to problem (3.22). Lastly, $(\underline{u}, \underline{v})$ and $(\overline{u}, \overline{v})$ are also attractive in the sense that

holds uniformly on $[0, T]\times\overline{\Omega}(0)$ for any initial conditions $(u_0(x), v_0(x))\geq, \not\equiv(0, 0)$ on $\overline{\Omega}(0)$.

At the last, in order to evaluate the effect of the periodically evolving domain on the viability of mistletoes, we present the following investigation, which results directly from Theorem 3.3 (i).

Theorem 3.12. If $\alpha(t, x)\equiv\alpha(t)$, $\delta(t, x)\equiv\delta(t)$ for all $t\in[0, T]$, then the following statements hold:

(i) if $\overline{\kappa^{-2}} = 1$, then we have $R^M_0 = \Re^M_0$;

(ii) if $\overline{\kappa^{-2}} > 1$, then we have $R^M_0 < \Re^M_0$;

(iii) if $\overline{\kappa^{-2}} < 1$, then we have $R^M_0 > \Re^M_0$,

where $\overline{\kappa^{-2}} = \frac{1}{T}\int^T_0\frac{1}{\kappa^2(t)}\mathrm{d}t$.

In fact, Theorem 3.12 reveals that $\overline{\kappa^{-2}}$ can be deemed to an index to forecast the impact of the periodic evolution of domain on the viability of mistletoes. If $\overline{\kappa^{-2}} = 1$, then the periodical domain evolution has no effect on the viability of mistletoes. However, if $\overline{\kappa^{-2}} > 1$, we guess that the domain evolution caused by the diffusion of mistletoes is not conducive to their survival and transmission. Lastly, if $\overline{\kappa^{-2}} < 1$, we find that mistletoes can survive well on the fixed and evolving domains, what is more, the domain evolution can promote diffusion of mistletoes and thus give mistletoes more space for transmission. In brief, the average expansion of the domain can enhance the viability of mistletoes, and vice versa.

4.   Numerical simulations and conclusions

In this section, we will implement some numerical simulations to test the previous theoretical results. In addition, to put more emphasis on the diffusive dynamics of mistletoes under the influence of birds, we give the following assumptions about the some coefficients.

To better survey the asymptotic behaviors of the solution of problem (1.4)–(1.5), we select different $\alpha$, $\delta$ and $\kappa(t)$.

Example 4.1 Fix $\alpha = 0.1$ and $\delta = 0.09905$. Choose different $\kappa(t)$:

(i) Let $\kappa(t) = 1$, then

It is easy to know from Figure 1 that $m$ rapidly decreases to 0, which agrees with Theorem 3.11 (i) that at length mistletoes on a fixed domain will go to extinction;

(ii) Set $\kappa(t) = e^{0.1(1-\cos(4t))}$, calculation obtains

and

It is shown in Figure 2 (a) that $m$ tends to a positive steady state. (b)–(c) implies that domain is periodically evolving. And it is line with Theorem 3.10 that mistletoes can coexist with birds on a periodically evolving domain, which also consists with Theorem 3.12 (iii).

Example 4.2. Fix $\alpha = 0.0845$ and $\delta = \frac{1}{12}$. Choose different $\kappa(t)$:

(i) Let $\kappa(t) = 1$, then

Observing Figure 3, we easily find that $m$ stabilizes a positive steady state, which is consistent with Theorem 3.11 (ii) that mistletoes and birds can coexist in a fixed domain;

(ii) Set $\kappa(t) = e^{0.1(\cos(4t)-1)}$, result is

and

Figure 4 (a) shows that $m$ decays to 0 eventually. (b)–(c) tells us that domain evolves periodically. And it agrees to Theorem 3.10 that mistletoes will be eradicated in a periodically evolving domain, which also consists with Theorem 3.12 (ii).

Acknowledgments

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions, which have expressly promoted the original manuscript. This research was partially supported by National Natural Science Foundation of China [11701243, 11901264] and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology.

Conflict of interest

The authors declare there is no conflict of interest.