Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response

In any reaction-diffusion system of predator-prey models, the population densities of species are determined by the interactions between them, together with the influences from the spatial environments surrounding them. Generally, the prey species would die out when their birth rate is too low, the habitat size is too small, the predator grows too fast, or the predation pressure is too high. To save the endangered prey species, some human interference is useful, such as creating a protection zone where the prey could cross the boundary freely but the predator is prohibited from entering. This paper studies the existence of positive steady states to a predator-prey model with reaction-diffusion terms, Beddington-DeAngelis type functional response and non-flux boundary conditions. It is shown that there is a threshold value $\theta_0$ which characterizes the refuge ability of prey such that the positivity of prey population can be ensured if either the prey's birth rate satisfies $\theta\geq\theta_0$ (no matter how large the predator's growth rate is) or the predator's growth rate satisfies $\mu\le 0$, while a protection zone $\Omega_0$ is necessary for such positive solutions if $\theta<\theta_0$ with $\mu>0$ properly large. The more interesting finding is that there is another threshold value $\theta^*=\theta^*(\mu,\Omega_0)<\theta_0$, such that the positive solutions do exist for all $\theta\in(\theta^*,\theta_0)$. Letting $\mu\rightarrow\infty$, we get the third threshold value $\theta_1=\theta_1(\Omega_0)$ such that if $\theta>\theta_1(\Omega_0)$, prey species could survive no matter how large the predator's growth rate is. In addition, we get the fourth threshold value $\theta_*$ for negative $\mu$ such that the system admits positive steady states if and only if $\theta>\theta_*$.Comment: 18 page

Logistic approximations and their consequences to bifurcations patterns and long-run dynamics

On infinitesimally short time interval various processes contributing to population change tend to operate independently so that we can simply add their contributions (Metz and Diekmann (1986)). This is one of the cornerstones for differential equations modeling in general. Complicated models for processes interacting in a complex manner may be built up, and not only in population dynamics. The principle holds as long as the various contributions are taken into account exactly. In this paper we discuss commonly used approximations that may lead to dependency terms affecting the long run qualitative behavior of the involved equations. We prove that these terms do not produce such effects in the simplest and most interesting biological case, but the general case is left open.Comment: 1 figur

Number and Stability of Relaxation Oscillations for Predator-Prey Systems with Small Death Rates

We consider planar systems of predator-prey models with small predator death rate $\epsilon>0$. Using geometric singular perturbation theory and Floquet theory, we derive characteristic functions that determines the location and the stability of relaxation oscillations as $\epsilon\to 0$. When the prey-isocline has a single interior local extremum, we prove that the system has a unique nontrivial periodic orbit, which forms a relaxation oscillation. For some systems with prey-isocline possessing two interior local extrema, we show that either the positive equilibrium is globally stable, or the system has exact two periodic orbits. In particular, for a predator-prey model with the Holling type IV functional response we derive a threshold value of the carrying capacity that separates these two outcomes. This result supports the so-called paradox of enrichment.Comment: 32 pages, 13 figure

Prey cannibalism alters the dynamics of Holling-Tanner type predator-prey models

Cannibalism, which is the act of killing and at least partial consumption of conspecifics, is ubiquitous in nature. Mathematical models have considered cannibalism in the predator primarily, and show that predator cannibalism in two species ODE models provides a strong stabilizing effect. There is strong ecological evidence that cannibalism exists among prey as well, yet this phenomenon has been much less investigated. In the current manuscript, we investigate both the ODE and spatially explicit forms of a Holling-Tanner model, with ratio dependent functional response. We show that cannibalism in the predator provides a stabilizing influence as expected. However, when cannibalism in the prey is considered, we show that it cannot stabilise the unstable interior equilibrium in the ODE case, but can destabilise the stable interior equilibrium. In the spatially explicit case, we show that in certain parameter regime, prey cannibalism can lead to pattern forming Turing dynamics, which is an impossibility without it. Lastly we consider a stochastic prey cannibalism rate, and find that it can alter both spatial patterns, as well as limit cycle dynamics

About reaction-diffusion systems involving the Holling-type II and the Beddington-DeAngelis functional responses for predator-prey models

We consider in this paper a microscopic model (that is, a system of three reaction-diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to $0$ towards the solutions of a reaction-cross diffusion system of predator-prey type involving a Holling-type II or Beddington-DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington-DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion

Modeling Boyciana-fish-human Interaction with Partial Differential Algebraic Equations

With human social behaviors influence, some boyciana-fish reaction-diffusion system coupled with elliptic human distribution equation is considered. Firstly, under homogeneous Neumann boundary conditions and ratio-dependent functional response the system can be described as a nonlinear partial differential algebraic equations (PDAEs) and the corresponding linearized system is discussed with singular system theorem. In what follows we discuss the elliptic subsystem and show that the three kinds of nonnegative are corresponded to three different human interference conditions: human free, overdevelopment and regular human activity. Next we examine the system persistence properties: absorbtion region and the stability of positive steady states of three systems. And the diffusion-driven unstable property is also discussed. Moreover, we propose some energy estimation discussion to reveal the dynamic property among the boyciana-fish-human interaction systems.Finally, using the realistic data collected in the past fourteen years, by PDAEs model parameter optimization, we carry out some predicted results about wetland boyciana population. The applicability of the proposed approaches are confirmed analytically and are evaluated in numerical simulations.Comment: 24 pages, 12 figure

Uniform persistence in a prey-predator model with a diseased predator

Following the well-extablished mathematical approach to persistence and its recent developments we give a rigorous theoretical explanation to the numerical results obtained for a certain prey-predator model with functional response of Holling type II equipped with an infectious disease in the predator population. The proof relies on some repelling conditions that can be applied in an iterative way on a suitable decomposition of the boundary. A full stability analysis is developed, showing how the "invasion condition" for the disease is derived. Some counterexamples and possible further investigations are discussed.Comment: 17 pages, 2 figure

Conditions for Permanence and Ergodicity of Certain Stochastic Predator-Prey Models

This work derives sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with Beddington-DeAngelis functional response. The conditions obtained in fact are very close to the necessary conditions. Both non-degenerate and degenerate diffusions are considered. One of the distinctive features of our results is that our results enables characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to existing literature and related matters to other stochastic predator-prey models are also given.Comment: Journal of Applied Probabilit

Hopf bifurcation of the Michaelis-Menten type ratio-dependent predator-prey model with age structure

This paper is devoted to the study of a predator-prey model with predator-age structure that involves Michaelis-Menten type ratio-dependent functional response. We study some dynamical properties of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain. The existence of Hopf bifurcation is established by regarding the biological maturation period $\tau$ as the bifurcation parameter. The computer simulations and sensitivity analysis on parameters are also performed to illustrate the conclusions

Dynamical behaviour of an ecological system with Beddington-DeAngelis functional response

The objective of this paper is to study the dynamical behaviour systematically of an ecological system with Beddington-DeAngelis functional response which avoids the criticism occurred in the case of ratio-dependent functional response at the low population density of both the species. The essential mathematical features of the present model have been analyzed thoroughly in terms of the local and the global stability and the bifurcations arising in some selected situations as well. The threshold values for some parameters indicating the feasibility and the stability conditions of some equilibria are also determined. We show that the dynamics outcome of the interaction among the species are much sensitive to the system parameters and initial population volume. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions are also derived with the use of both the normal form and the central manifold theory (cf. Carr \cite{Carr}). Numerical illustrations are performed finally in order to validate the applicability of the model under consideration.Comment: 21 pages, 8 figure