Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle

1.   Introduction

Spatial propagation dynamics of parabolic type systems has been widely investigated in literature, and two important indices on spatial propagation are minimal wave speed and spreading speed. Here, the minimal wave speed is the threshold on the existence of specific traveling wave solutions and the spreading speed of a nonnegative function $u(x,t), x\in\mathbb R, t>0$ is defined as follows [1].

Definition 1.1. Assume that $u(x,t)$ is a nonnegative function for $x\in \mathbb{R}, t>0$. Then $\widehat{c}$ is called the spreading speed of $u(x,t)$ if

a): $\lim_{t\rightarrow \infty} \sup_{|x|>(\widehat{c}+\epsilon)t}u(x,t) = 0$ for any given $\epsilon >0$;

b): $\liminf_{t\rightarrow \infty} \inf_{|x|<(\widehat{c}-\epsilon)t}u(x,t)>0$ for any given $\epsilon \in (0,\widehat{c})$.

From the viewpoint of mathematical biology, the above speed characterizes the spatial expansion of the individuals [28,34]. In the past decades, some important results on these two thresholds have been established for monotone semiflows, see [12,18,26,37,38] and a survey paper by Zhao [43]. When some special cooperative systems are concerned, it has been proven that all components governed by a system have the same spreading speed that is also the minimal wave speed of traveling wave solutions [18,26,38]. At the same time, it has been shown that different components may have different spreading speeds in several noncooperative systems [19,20,21,22,30,33], and at least the spreading speed of one species equals to the minimal wave speed of traveling wave solutions.

Recently, Li et al. [17] investigated the following nonmonotone system

where $x\in\mathbb{R}, t>0,$ $(u_1,u_2)\in \mathbb{R}^2,$ $r_1,r_2, d_1,d_2$ are positive constants, $F_1,F_2$ are defined by

with constants $b_1\ge 0,b_2 \ge 0, a_1\ge 0, a_2\ge 0,\tau > 0$ and

In this system, $[J_1*u_1](x,t),[J_2*u_2](x,t)$ reflecting the spatial dispersal indicate the long distance effect and nonadjacent contact among individuals [4,13,14], and are defined by

where $J_i,i = 1,2,$ play the role of probability kernel functions about the random walk of individuals and satisfy the following assumptions:

(J1): $J_i$ is nonnegative and continuous for each $i = 1,2$;

(J2): for any $\lambda \in \mathbb{R},$ $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy < \infty, i = 1,2;$

(J3): $\int_{\mathbb{R}}J_i(y)dy = 1$, $J_i(y) = J_i(-y), y\in \mathbb{R}, i = 1,2.$

Clearly, (1) is a predator-prey system in population dynamics. In Li et al. [17], Yu and Yuan [41], Zhang et al. [42], the authors investigated its traveling wave solutions connecting $(0,0)$ with the positive steady state, which reflect the process that these two species invade a new habitat from the viewpoint of biology invasion. In particular, Li et al. [17] obtained the minimal wave speed defined by $c^{\ast } = \max \{c_{1}^{\ast },c_{2}^{\ast }\}$ with

From the viewpoint of initial value problem, let any fixed time be the initial time, the traveling wave solutions in [17,41,42] indicate the initial size of habitat of both species is infinite, which contradicts to some natural phenomena because the initial invasion often begins in finite domain. The purpose of this paper is to explore the dynamics when the initial habitats of two invaders are finite and investigate the long time behavior of

in which $\phi_i(x,s)$ satisfies

(Ⅰ): For $i = 1,2,$ $\phi_i(x,s),x\in \mathbb R,s\in [-\tau, 0],$ is nonnegative, bounded and continuous such that

for some $L>0, x_i\in \mathbb R.$ Moreover, they satisfy

Since (2) involves delay effect of intraspecific competition if $b_1+b_2 >0$, it does not satisfy the comparison principle of classical predator-prey systems or monotone semiflows [40]. Therefore, the spreading speeds can not be investigated by the abstract results mentioned above. In this paper, we shall estimate the spreading speeds of these two species. By constructing proper auxiliary equations and functions, we confirm that either the predator or the prey invades the new habitat at the rough speed $c^{\ast }$ while the spreading speed of the other species may be smaller than $c^*.$

2.   Main results

In this section, we shall give and prove the main results on (2). Before giving the main results, we first define some positive constants as follows

in which the existence and uniqueness of $\lambda_2 >0$ is due to (J2)-(J3) and the convex of $d_{2}\left[ \int_{\mathbb{R} }J_{2}(y)e^{\lambda y}dy-1\right] -c \lambda+r_{2}$ in $\lambda >0$ for every $c>0.$ Using these constants, we present the following conclusion.

Theorem 2.1. Assume that the mild solution $(u_1(x,t), u_2(x,t))$ is defined by (2). Then it satisfies

Moreover, $(u_1(x,t), u_2(x,t))$ satisfies the following properties.

(1): If $c_1>c_2$ is true, then $c^*$ is the spreading speed of $u_1(x,t)$ while the spreading speed of $u_2(x,t)$ is not larger than $c_2.$

(2): Further suppose that

If $c_1^*\le c_2^*$ is true, then $c^*$ is the spreading speed of $u_2(x,t)$ while the spreading speed of $u_1(x,t)$ is not larger than $c_1^*.$

We now prove the above theorem by several lemmas. Let $X$ be the Banach space of uniformly continuous and bounded functions equipped with supremum norm. For each $i\in\{1,2\},$ we see that $d_i[J_i\ast v](x):X\to X$ is a bounded linear operator by (J1), so

generates a positive $C_0$ semigroup $T_i(t): X\to X, t\ge 0$ (see [39,Lemma 3.1]), and the mild solution of the above initial value problem is denoted as

for any $0\le s <t <\infty.$ Moreover, for any given kernel function $J$ satisfying (J1)-(J3), it also generates a positive $C_0$ semigroup $T(t): X\to X, t\ge 0$. Consider the following initial value problem

where $J$ satisfies (J1)-(J3), $d >0$ and $r>0$ are constants. Also define

By Jin and Zhao [15], we have the following conclusion.

Lemma 2.2. Assume that $0\le \chi(x)\le 1.$ Then (5) admits a solution $u(\cdot, t)\in X$ for all $t>0,$ which also satisfies

for $x\in\mathbb R, 0\le s <t <\infty.$ If $w(\cdot,t)\in X, t\ge 0$ is nonnegative and bounded such that

or

for $x\in\mathbb R, 0\le s <t <\infty,$ then

If $\chi(x)$ has nonempty support, then for any $c<c',$ we have

If $\chi(x)$ has compact support, then

Remark 1. By the positivity of the semigroup, if

for $i\in \{1,2,3\},$ then

On the existence of mild solution of (1), we have the following result.

Lemma 2.3. The positive mild solution $(u_1(\cdot,t), u_2(\cdot,t))\in X^2$ exists for all $t>0$ and satisfies (3).

Proof. The local existence is evident by the theory of abstract functional differential equations [27], here the mild solution is defined by

for $0\le \theta <t <T$ with some $T\in (0,\infty].$ If $T = \infty,$ then the global existence is true.

Further by the quasipositivity in $u_1F_1,u_2F_2,$ we see the mild solution is nonnegative. If $u_1(x,t)$ only exists for $t\in [0,T)$ with bounded $T$ such that

then

for $0\le \theta <t <T$, and the comparison principle (Lemma 2.2) implies

A contradiction occurs. The proof is complete by similar discussion on $u_2(x,t).$

To continue the discussion, we investigate the following scalar equation

where $x\in \mathbb{R},t>0, s\in [-\tau, 0],$ $J$ satisfies (J1)-(J3), $d>0,r>0, b\ge 0,$

Furthermore, $\nu (x,s)\ge 0$ is uniformly continuous and bounded. Evidently, the global existence of mild solution of (7) is true by Lemma 2.3.

Lemma 2.4. Assume that $v(x,t)$ is the mild solution defined by (7). If $\nu (x,0)$ admits nonempty compact support such that $0\le \nu (x,0) \le 1, x\in \mathbb R$, then its spreading speed is $c'$ defined by (6).

Proof. We now prove it by the idea in Liu and Pan [25]. If

then the result is clear by Lemma 2.2. Otherwise, the positivity implies that

for any $0\le s <t<\infty,$ then Lemma 2.2 implies $v(x,t)\le 1$ and

For any fixed $c < c',$ it suffices to prove that

For the purpose, we select $\epsilon >0$ such that

and $\tau ' \in (0,\tau)$ such that

If $b \int_{-\tau}^{-\tau '}v(x, t+s)d\eta (s) \le 2\epsilon,$ then

When $b \int_{-\tau}^{-\tau '}v(x, t+s)d\eta (s) > 2\epsilon,$ there exists $s_0 \in [-\tau, -\tau ']$ such that

and the uniform continuity implies

for some $\sigma >0.$ Consider the initial value problem

where $\underline{\nu} (x)$ is a continuous function satisfying

(1): $\underline{\nu} (x) = \frac{\epsilon}{b\tau}, |x| \le \sigma /2;$

(2): $\underline{\nu} (x) = 0, |x| \ge \sigma ;$

(3): $\underline{\nu} (x)$ is even and decreasing in $x\in [\sigma /2, \sigma ].$

By the positivity of $T(t)$ and the property of continuous functions, we see that $\underline{v}(0,t)$ is positive in $t>0,$ and there exists $\rho >0$ such that $\underline{v}(0,t) > \rho, t\in [\tau ', \tau]$ and so

That is, $v(x,t)$ satisfies

for all $0\le s <t <\infty.$ The proof is complete by Lemma 2.2.

Lemma 2.5. If $c_1>c_2$ is true, then $c^*$ is the spreading speed of $u_1(x,t)$ while the spreading speed of $u_2(x,t)$ is not larger than $c_2.$

Proof. By (3), $u_2$ satisfies

for $0\le \theta <t <\infty,$ so Lemma 2.2 implies that the spreading speed of $u_2(x,t)$ is not larger than $c_2$, which also leads to

Again by (3), we see that

for all $0\le \theta <t <\infty,$ and Lemma 2.4 or its proof implies

We now verify that $c^*$ is the spreading speed of $u_1(x,t).$ Because of Lemma 2.2 and

for all $0\le \theta <t <\infty,$ it suffices to confirm that

for any given $c <c^*.$ We now fix $3 c > c_2+2c_1$ and $\epsilon >0$ such that

By (9) and (10), there exists $T>0$ such that

and

which implies that there exists $M>0$ depending on $\epsilon$ such that

for any $T\le \theta <t .$ Clearly, the spreading speed is not less than that of

by repeating the proof of Lemma 2.4. The proof is complete.

Lemma 2.6. Assume that (4) is true. If $c_1^*\le c_2^*$ is true, then $c^*$ is the spreading speed of $u_2(x,t)$ while the spreading speed of $u_1(x,t)$ is not larger than $c_1^*.$

Proof. By the positivity of (3), $u_1(x,t)$ satisfies

for any $0\le s < t ,$ the spreading speed of $u_1(x,t)$ is not larger than $c_1^*$ by Lemma 2.2.

On the other hand, $u_2(x,t)$ satisfies

for any $0\le s < t ,$ and Lemma 2.4 implies that the spreading speed of $u_2(x,t)$ is not less than $c^*.$ Now, we shall prove that

for any fixed $c > c^*.$ By the positivity, we obtain

and the result is true if the spreading speed of the following equation is $c^*$

where $\underline{u}_{2}(x,0) = u_2(x,0),$ $\overline{u}_{1}(x,s) = \phi_1(x,s), s\le 0,$ and $\overline{u}_{1}(x,t), t>0$ is defined by

The main reason why the above claim is true is that the above equation (12) is monotone and admits comparison principle.

By Remark 1 and (4), we see that there exists $T_1>0$ such that

In fact, let $\widehat{u}_1(x,t) = e^{\lambda_2(x+c_2 t +T_1)}$ or $e^{\lambda_2(-x+c_2 t +T_1)},$ then

and

if $\widehat{u}_1(x,t) = 1.$ Further select $T_2 \ge T_1$ such that

and

Then Remark 1 implies that

because $\widehat{u}_{2}(x,t) = e^{\lambda_2(x+c_2 t +T_2)} \left( e^{\lambda_2(-x+c_2 t +T_2)}, 1+a_2\right)$ implies

The proof is complete.

3.   Discussion

The propagation dynamics of predator-prey systems has important ecology background, one typical case is the evolution of insect herbivores and lupins on Mount St Helens, see [11,29]. Another related topic is the asymptotic spreading in epidemic models because of the similar monotone conditions. In literature, much attention has been paid to the traveling wave solutions since the work of Dunbar [10]. However, there are a few results on asymptotic spreading of predator-prey systems, see part results by Ducrot [7,8], Ducrot et al. [9], Lin [19], Lin et al. [23], Pan [30,32], Wang and Zhang [36].

The model in this paper admits nonlocal dispersal and time delays, the mechanism has significant biological reasons and other backgrounds [4,5,13,14,28], and the monotone case has been widely studied, see some recent works by [2,3,6]. Similar to that in [16,24], the model in this paper is not a monotone system [35] and time delay is not small enough. When the diffusion is of the classical Ficker type in (1), Pan [31] studied its minimal wave speed of traveling wave solutions connecting trivial steady state with the positive one.

In this paper, we show that $c^*$ may be the spreading speed of $u_1$ or $u_2,$ which is the minimal wave speed in [17]. It is possible to study the asymptotic spreading of the model in [31] by our idea. Both thresholds in this paper and [17] formulate that two species invade a new habitat. From our results, we see that the predator and the prey may have different spreading speeds, but it is difficult to estimate these spreading speeds. To answer these questions, more properties on the nonlocal operator and delayed systems are necessary. We shall further investigate these questions in our future research.