Abstract
Allee effect is one of the important factors in ecology, and taking it into account can cause significant impacts in the system dynamics. In this paper, we study the dynamics of a two-prey one-predator system, where the growth of both prey populations is subject to Allee effects, and there is a direct competition between the two-prey species having a common predator. Two discrete time delays τ1 and τ2 are incorporated into the model to represent reaction time of predators. Sufficient conditions for local stability of positive interior equilibrium and existence of Hopf bifurcations in terms of threshold parameters τ1∗ and τ2∗ are obtained. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Sensitivity analysis to evaluate the uncertainty of the state variables to small changes in the Allee parameters is also investigated. Presence of Allee effect and time delays in the model increases the complexity of the model and enriches the dynamics of the system. Some numerical simulations are provided to illustrate the effectiveness of the theoretical results. The model is highly sensitive to small changes in Allee parameters at the early stages and with low population densities, and this sensitivity decreases with time.
Highlights
Suppose that N(t) is the size of prey population and P(t) is the size of the predator population at time t, the Lotka–Volterra model is given by the following equations: dN(t) dt
We extend the work in [26] and study the dynamics of a two-prey one-predator system, where the growth of both prey populations is subject to Allee effects, and there exists a direct competition between the two-prey species having a common predator
We established the two-prey one-predator mode with time delays and a weak Allee effect in the preys’ growth functions, where there is a direct competition between prey populations
Summary
Many studies have been done on multispecies prey-predator systems, including local and global bifurcations and different types of chaos (see, e.g., [26,27,28,29]). Takeuchi and Adachi [28] considered two preys with logistic growth rates and an exponential functional response, where the predator survives on two-prey populations. We incorporate Allee effects in the growth functions of the two-prey populations, and there exists a direct competition between the two-prey species having a common predator. While the interaction between the second species of prey and predator is assumed to be governed by Holling type II (cyrtoid functional) δy(t)z(t)/(1 + cy(t)), response indicates that it is a hard-to-capture prey compared to the first species (see Figure 2). To investigate the role of time delay and Allee effect on the dynamics of the system, we first discuss the boundedness and positivity of the solutions of system (3) with the given positive initial conditions (4). Due to the positivity of z and since the parametric condition exists for ρ, the differential inequality is bounded above, such that (dW/dt) ≤ M − ρW, i.e., there exists N where 0 < W(t) < N for all t > T, which implies the boundedness of z, such that limt⟶∞supz(t) ≤ N
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