Dynamic behaviour of a delayed predator–prey model with harvesting

Abstract

In this paper, we analyze the dynamics of a delayed predator–prey system in the presence of harvesting. This is a modified version of the Leslie–Gower and Holling-type II scheme. The main result is given in terms of local stability, global stability, influence of harvesting and bifurcation. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by using the normal form method and center manifold theorem.

Introduction

It is well known that the diversity of biological phenomena determines the complexity of biological and mathematical models. While investigating biological phenomena, there are many factors which affect dynamical properties of biological and mathematical models. One of the familiar nonlinear factors is functional response. In population dynamics, a functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes [23]. Holling [10] suggested three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka–Volterra system more realistic. Leslie [17] introduced a predator–prey model where the carrying capacity of the predator environment is proportional to the number of prey. In fact, he stresses the fact that there are upper limits to the rate of increase of both prey and predator, which are not recognized in the Lotka–Volterra model.

The Leslie–Gower formulation is based on the assumption that reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food. Indeed, Leslie [17] introduced a predator–prey model where the carrying capacity of the predator’s environment is proportional to the number of prey. This interesting formulation for the predator dynamics has been discussed by Leslie and Gower [16] and Pielou [19].

It is well known that the harvesting has a strong impact on the dynamics of a model. In a harvesting model, the aim is to determine how much we can harvest without altering dangerously the harvested population. The MSY is a simple way to manage resources taking into consideration that over exploiting resources lead to a loss in productivity. But the main problem of the MSY is its economical irrelevance. For example, it ignores the fact that if a species is harvested such that its population decreases to a certain level, then the cost of harvesting can become exorbitant. Actually, many population harvesting activities have not been managed at all. If it had been managed, it was mainly using the MSY, which often gave rise to critical situations. Truly speaking the management of renewable resources is complicated and constructing accurate mathematical models is even more complicated. Therefore, the best we can do is to look for analyzable models that describe the effect of harvesting on populations. The effect of harvesting on the dynamics of predator–prey systems can be found in [5], [6], [12], [13], [21].

The predator–prey food chain model with harvesting is generally described as:

$$\begin{array}{c}\dot{x}(t)=\mathit{ax}(t)-\varphi (x(t))y(t)-h_1\text{,}\hfill \\ \dot{y}(t)=-\mathit{by}(t)+\gamma \varphi (x(t))y(t)-h_2\text{,}\hfill \end{array}$$

where φ(x) is the functional response of the predator y(t). In this paper, we consider a predator–prey model with harvesting which incorporates a modified version of the Leslie–Gower model (see [1], [2], [14], [18], [22]). The model is as follows:

$$\begin{array}{c}\dot{x}(t)=\left(r_1-b_{1}x(t)-\frac{a_{1}y(t)}{x(t)+k_1}\right)x(t)-h_1\text{,}\hfill \\ \dot{y}(t)=\left(r_2-\frac{a_{2}y(t)}{x(t)+k_2}\right)y(t)-h_2\text{,}\hfill \end{array}$$

where x and y represent the population densities of prey and predator respectively at time t; b₁, r_i, a_i, k_i (i = 1, 2) are model parameters assuming only positive values; r₁ is the growth rate of the prey population x, b₁ measures the strength of competition among individuals of species x, a₁ is the maximum value which per capita reduction rate of the prey population x can attain, k₁ (respectively, k₂) measures the extent to which environment provides protection to prey x (respectively, to predator y), r₂ describes the growth rates of the predator y, and a₂ has a similar meaning to a₁.

From the point of view of human need, the exploitation of biological resources and the harvesting of populations are commonly practiced in fishery, forestry and wildlife management. However, human need is not invariable for a long time. In detail, human need increases as biological resources become abundant, while human need decreases as biological resource is exiguous. Thus we focus on varying harvesting rate. Also we consider a negative feedback τ > 0, of the predator density.

Thus we consider the model as:

$$\begin{array}{c}\frac{\mathit{dx}}{\mathit{dt}}=r_{1}x-b_1{x}^2-\frac{a_1\mathit{xy}}{x+k_1}-c_{1}x\text{,}\hfill \\ \frac{\mathit{dy}}{\mathit{dt}}=y\left(r_2-\frac{a_{2}y(t-\tau )}{x(t-\tau )+k_2}\right)-c_{2}y\text{,}\hfill \end{array}$$

wherex(0) = x₀ ⩾ 0, y(0) = y₀ ⩾ 0, c₁ and c₂ are the harvesting coefficients of prey and predator respectively. For θ ∈ [−τ, 0], we use the notation

$$x_t(\theta )=x(t+\theta )\text{.}$$

Then the initial conditions for this system take the form

$$\begin{array}{c}{x}_0(\theta )={\varphi }_1(\theta )\text{,}\hfill \\ y_0(\theta )={\varphi }_2(\theta )\text{,}\hfill \end{array}$$

for all θ ∈ [−τ, 0], where $({\varphi }_1\text{,}{\varphi }_2)\in C([-\tau \text{,}0]\text{,}{R}_{+}^2)$, x(0) = ϕ₁ > 0 and y(0) = ϕ₂ > 0. System (1.3) assumes that the change rate of the predators depends on the number of prey and predators present at some previous time. In general, the delay differential equations exhibit much more complicated dynamics than ordinary differential equations (see [15], [7], [8], [3], [20]).

The object of this paper is to study the combined effects of harvesting and delay on the dynamics of a Leslie–Gower prey–predator model. The reason for choosing this model is that, since we know the dynamics of the system, it will be better for us to determine the effects of delay and harvesting.