Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting

Highlights

• A predator-prey model with prey protection zone and predator harvesting is investigated.

• Existence, positivity, uniqueness of solution is shown.

• The existence and uniqueness of the positive homogeneous steady state is proved.

• Global stability of the boundary equilibrium steady state is investigated under some value of model parameters.

• The existence of Hopf, Turing, Turing-Hopf bifurcations is investigated.

• The stability of the periodic solutions are studied using the normal form.

• Some numerical simulations are used for discussing the impact of the predator harvesting on the spatiotemporal behavior of solution.

Abstract

In this paper, a diffusive predator-prey model subject to the zero flux boundary conditions is considered, in which the prey population exhibits social behavior and the harvesting functional of the predator population is assumed to be considered in a quadratic form. The existence of a positive solution and its bounders is investigated. The global stability of the semi trivial constant equilibrium state is established. Concerning the non trivial equilibrium state, the local stability, Hopf bifurcation, diffusion driven instability, Turing-Hopf bifurcation are investigated. The direction and the stability of Hopf bifurcation relying on the system parameters is derived. Some numerical simulations are used to extend the analytical results and show the occurrence of the homogeneous and non homogeneous periodic solutions. Further the effect of the rivalry rate on the dynamical behavior of the studied species.

Introduction

The ecosystems becomes one of the most important tools to understand the interaction between species [4], [30], [33], [47]. The predator-prey model is one of the most used model in ecosystems. It has been considered as the dominant theme in mathematical ecology for a long time [4], [30], [33], [47]. The big part of the recent papers in mathematic ecology are devoted to study the interaction between the predator and the prey for the subject of conserving the studied species. In terms of modeling, the total dynamics of a predator-prey system can be affected by many factors such as predator efficiency of their attack, birth, death, mature delay, so on. It is well known that the functional response is the crucial component to describe the relationship between the prey and the predator populations. In the last few decades, many functional responses appears to study and describe a particular relationship between the studied species. It is the main reason for having many functional responses such as Holling $I-IV$ functional response [27], Beddington-DeAngelis functional response [9], ratio-dependent functional response, Crowley-Martin functional response [45], [59], Hassell-Varley functional response, so on.

Now, introducing a prototypical predator-prey model with the following structure

$$\{\begin{array}{c}{\displaystyle \frac{dU}{\partial \tau }=Uf\left(U\right)-g(U,V),}\hfill \\ {\displaystyle \frac{dV}{\partial \tau }=\tilde{\beta }(-\tilde{m}V+g(U,V)),}\hfill \end{array}$$

where U, V are respectively the populations sizes of the prey and the predator at the time t, f(U) is the net growth rate for the prey population in the absence of the predator population, $0<\tilde{\beta }<1$ is the prey consuming rate of a predator to a prey, $\beta \tilde{m}$ is the natural death rate of the predator.

In the real world, the prey population has a fast reproduction comparing with the predator population. It is wise to choose a reproduction functional f(U) to model the crowding effect of the prey. The most known one is the logistic form:

$$f\left(U\right)=r(1-\frac{U}{K})$$

where r is the net growth rate of the prey population; K is the carrying capacity of the space for the prey population. Besides, the functional g(U, V) represents the interaction functional. As we said, it models the quality of interaction between the prey and the predator.

In nature, an important part of savanna preys lives in packs such as zebras, buffalos and elephants. This behavior can affect the availability of some prey to the predator. This behavior can make a huge benefit to the prey population, where the pack can give an improvement to the protection of the prey from the predator. It can be seen in the partition of the herd where the weakest prey will be inserted in the inside herd and the strongest ones on the outer corridor of the herd. Recently, this case of the interaction between the prey and the predator populations has been modeled using the functional response $g(U,V)=a\sqrt{U}$ (where a is real positive constant) due to the paper presented by Ventorino et al [2]. The main reason for proposing such as functional response is based on the non availability of the inside prey of the herd to the predator. More precisely, the predator hunt only on the confines of the prey pack. The number of consumed prey by one predator will be the proportional to the density on the borders of the herd. This case of prey behavior entice many researchers. We refer readers to the papers [2], [8], [12], [14], [15], [24], [46], [48]. In this paper, we will investigate with the functional response $g(U,V)=a\sqrt{U},$ the system (1.1) becomes

$$\{\begin{array}{c}{\displaystyle \frac{dU}{\partial \tau }=rU(1-\frac{U}{K})-a\sqrt{U}V,}\hfill \\ {\displaystyle \frac{dV}{\partial \tau }=\tilde{\beta }V(-\tilde{m}+a\sqrt{U}),}\hfill \end{array}$$

On the other hand the case of herd behavior for the prey population can affect the availability of the prey to the predator, and using the fact that the most part of the predator population put their focus on hunting the weakest one (the weakest prey located in inside of the pack) which can make perturbation on the predator population, where will push the predator to fight with each other for a small quantity of resources, this behavior is known by harvesting. Also, it can be seen on the competition between the predators for the hunting zones for the exclusivity of reproduction. The most famous predator that can exhibit this behavior is the lions, where the solitary males enter into hunting zone of other lions and fight the their leader for the exclusivity of reproduction. The harvest has a strong influence on the dynamic evolution of a population, this phenomenon is common in the fields of forestry, fishing and wildlife management. It is well know that these predators in a huge danger due to the human intervention and disappearance of many zones of living for these living being. The interest of this paper is to study the effect of the herd behavior and the predator harvesting in the existence of the considered species. During the few last years, several forms of harvest have been proposed and studied, mainly consisting of constant harvest [31], [60], [61], proportional harvesting [29] and nonlinear harvesting [25], [59]. Here, we are instead to investigate a model that considers a quadratic harvesting which can be modeled by ($\tilde{\beta }\gamma V^2$). This case of predator has been widely investigates we cite for instance the papers [1], [3], [4], [5], [52], [53]. Based on the best knowledge there is no results on its effect on the interaction predator-prey in the presence of herd behavior where in this case the predator has two difficulties, the hunting from the herd, which is dangerous and protecting the zone of hunting from a predator of it gender. Further, in [40], it has been given a comparative analysis between the linear mortality and quadratic one, where two models have been investigated, but in the real world, the predator can exhibits the two types of mortality, where the previous example (lions example) is the simplest one that can hold the two types of mortality. The harvesting can hold for the smaller lions and the bigger males, and the natural mortality can insured for the females, where it has been noticed that in the savanna, the males never attacks their females (also in [56] it has been considered that the predator exhibits quadratic mortality only). Based on the above discussion we formulate the following system

$$\{\begin{array}{c}{\displaystyle \frac{dU}{\partial \tau }=rU(1-\frac{U}{K})-a\sqrt{U}V,}\hfill \\ {\displaystyle \frac{dV}{\partial \tau }=\tilde{\beta }V(-\tilde{m}+a\sqrt{U}-\gamma V),}\hfill \end{array}$$

Besides, the predator and the prey are in different spatial locations, there exist many reasons for moving such as search for resources, currents and turbulent diffusion. The movement of the prey and predator can be modeled by the presence of spatial diffusion on the system (1.3). The biological reaction diffusion systems becomes the one of the most dominant themes in mathematical biology we give as example the papers [38], [41], [42], [43], [55], [56], [57]. Naturally, the presence of spatial diffusion leads to the following predator-prey model

$$\{\begin{array}{cc}{\displaystyle \frac{\partial U}{\partial \tau }=rU(1-\frac{U}{K})-a\sqrt{U}V+d_1\Delta U,}\hfill & x\in \Omega ,\tau >0,\hfill \\ {\displaystyle \frac{\partial V}{\partial \tau }=\tilde{\beta }V(-\tilde{m}+a\sqrt{U}-\gamma V)+d_2\Delta V,}\hfill & x\in \Omega ,\tau >0,\hfill \\ {\displaystyle \frac{\partial U}{\partial \overrightarrow{n}}=\frac{\partial V}{\partial \overrightarrow{n}}=0,}\hfill & x\in \partial \Omega ,\tau >0,\hfill \\ U(x,0)=U_0\left(x\right)\ge 0,V(x,0)=V_0\left(x\right)\ge 0,\hfill & x\in \Omega ,\hfill \end{array}$$

where Ω is a bounded domain in ${\mathbb{R}}^n,n\ge 1$ with a smooth boundary and $\overrightarrow{n}$ is the unit outward unit normal vector of the boundary ∂Ω. The homogeneous zero flux boundary conditions describes isolated patches of the two populations. For the goal of reducing the number of parameters we set the following change of variables

$$u=\frac{U}{k},v=\frac{a}{r\sqrt{k}}V,t=r\tau ,\beta =\frac{a\tilde{\beta }\sqrt{k}}{r},c=\gamma \frac{r}{a},m=\frac{\tilde{m}}{a\sqrt{k}}.$$

Then the system (1.4) becomes

$$\{\begin{array}{cc}{\displaystyle \frac{\partial u}{\partial t}=u(1-u)-\sqrt{u}v+d_1\Delta u,}\hfill & x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\partial v}{\partial t}=\beta v(-m+\sqrt{u}-cv)+d_2\Delta v,}\hfill & x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\partial u}{\partial \overrightarrow{n}}=\frac{\partial v}{\partial \overrightarrow{n}}=0,}\hfill & x\in \partial \Omega ,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0,\hfill & x\in \Omega ,\hfill \end{array}$$

The most interesting about the presence of predator harvesting and the spatial diffusion on predator prey model with herd behavior is its effect on the existence of species (more precisely the harvesting). For the reason of the novelty of the prey herd behavior it is interesting to study the effect of the variable c on the spatiotemporal behavior of solution of the system (1.5). In the environment, the predator harvesting affect hugely on the existence of species, where for the small competition between the predators (which means that the predator are compatible) this can affect negatively the prey population and positively the predator, and the opposite for the bigger competitions. More examples on predator-prey models with prey herd behavior see [14], [15], [16], [17], [18], [19], [20], [21], [22], [37] and for more reading we cite the papers [6], [7], [10], [11], [13], [23], [28], [32], [34], [36], [39], [44], [49], [50], [51], [54], [58], [59]

For the purpose of achieving the cited goals, we organize the paper in the following form

In Section 2, the existence of a positive solution and it bounders for the system (1.5) are proved. Section 3, is devoted to study the existence of the equilibrium states for the system (1.5). In Section 4, the global stability of the semi trivial equilibrium state (1,0) is successfully established under a suitable condition on the model parameters. In Section 5, the spatiotemporal dynamics of the system (1.5) near the unique non trivial equilibrium state (u*, v*) is investigated, where the existence of Hopf bifurcation is discussed. On the other hand, the presence of Turing driven instability, Turing-Hopf bifurcation have been also discussed. For the stability of the homogeneous and nonhomogeneous periodic solutions generated by the presence of Hopf bifurcation are studied using the normal form on the center of manifold. The obtained results were checked using numerical simulations. A discussion section ends the paper.