Ratio-dependent Predator Prey Model Research Articles

AN EFFICIENT AND ACCURATE NUMERICAL SCHEME FOR TURING INSTABILITY ON A PREDATOR–PREY MODEL

We present an efficient and accurate numerical method for solving a ratio-dependent predator–prey model with a Turing instability. The system is discretized by a finite difference method with a semi-implicit scheme which allows much larger time step sizes than those required by a standard explicit scheme. A proof is given for the positivity and boundedness of the numerical solutions depending only on the temporal, but not on the spatial step sizes. Finally, we perform numerical experiments demonstrating the robustness and accuracy of the numerical solution for the Turing instability. In particular, we show that the numerical nonconstant stationary solutions exist.

RATIO-DEPENDENT PREDATOR–PREY MODEL WITH STAGE STRUCTURE AND TIME DELAY

Ratio-dependent predator–prey models are favored by many animal ecologists recently as more suitable ones for predator–prey interactions where predation involves searching process. In this paper, a ratio-dependent predator–prey model with stage structure and time delay for prey is proposed and analyzed. In this model, we only consider the stage structure of immature and mature prey species and not consider the stage structure of predator species. We assume that the predator only feed on the mature prey and the time for prey from birth to maturity represented by a constant time delay. At first, we investigate the permanence and existence of the proposed model and sufficient conditions are derived. Then the global stability of the nonnegative equilibria are derived. We also get the sufficient criteria for stability switch of the positive equilibrium. Finally, some numerical simulations are carried out for supporting the analytic results.

Bifurcation analysis of a ratio-dependent prey–predator model with the Allee effect

There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee effect. We both consider the cases of a strong Allee effect (the population growth rate is negative at low species density) and the case of a weak Allee effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee effect results in a paradox of biological control.

Ratio-dependent predator–prey model of interacting population with delay effect

A system of delay differential equation is proposed to account the effect of delay in the predator–prey model of interacting population. In this article, the modified ratio-dependent Bazykin model with delay in predator equation has been considered. The essential mathematical features of the proposed model are analyzed with the help of equilibria, local and global stability analysis, and bifurcation theory. The parametric space under which the system enters into a Hopf-bifurcation has been investigated. Global stability results are obtained by constructing suitable Lyapunov functions. We derive the explicit formulae for determining the stability, direction, and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc., 350:4799–4838, 1998) for functional differential equations, the global existence of periodic solutions has been established. Our analytical findings are supported by numerical experiments. Biological implication of the analytical findings are discussed in the conclusion section.

Multiple Periodic Solutions of a Ratio‐Dependent Predator‐Prey Discrete Model

A delayed ratio‐dependent predator‐prey discrete‐time model with nonmonotone functional response is investigated in this paper. By using the continuation theorem of Mawhins coincidence degree theory, some new sufficient conditions are obtained for the existence of multiple positive periodic solutions of the discrete model. An example is given to illustrate the feasibility of the obtained result.

Qualitative analysis on a diffusive and ratio-dependent predator-prey model

Qualitative analysis on a diffusive and ratio-dependent predator-prey model

Positive periodic solutions for neutral delay ratio-dependent predator–prey model with Holling type III functional response

By using a continuation theorem based on coincidence degree theory, some new and interesting sufficient conditions are obtained for the existence of positive periodic solutions for neutral delay ratio-dependent predator–prey model with Holling type III functional response x ′ ( t ) = x ( t ) [ r ( t ) - a ( t ) x ( t - σ 1 ) - ρ x ′ ( t - σ 2 ) ] - c ( t ) x 2 ( t ) m 2 y 2 ( t ) + x 2 ( t ) y ( t ) , y ′ ( t ) = y ( t ) - d ( t ) + h ( t ) x 2 ( t - τ ( t ) ) m 2 y 2 ( t - τ ( t ) ) + x 2 ( t - τ ( t ) ) . An example is represented to illustrate the feasibility of our main results.

DYNAMICS OF A STOCHASTIC RATIO-DEPENDENT PREDATOR-PREY MODEL

In this paper, we consider a stochastic ratio-dependent predator-prey model. We firstly prove the existence, uniqueness and positivity of the solutions. Then, the boundedness of moments of population are studied. Finally, we show the upper-growth rates and exponential death rates of population under some conditions.

A stochastic ratio-dependent predator-prey model under regime switching

This article presents an investigation of asymptotic properties of a stochastic ratio-dependent predator-prey model under regime switching. Both the white and color noises are taken into account in our model. We obtain the global existence of positive unique solution of the stochastic model. And we show the solution is bounded in mean. Moreover, the sufficient conditions for persistence in mean, extinction are obtained.

Positive periodic solutions for a neutral delay ratio-dependent predator–prey model with a Holling type II functional response

By using a continuation theorem based on coincidence degree theory, some new and interesting sufficient conditions are obtained for the existence of positive periodic solutions for a neutral delay ratio-dependent predator–prey model with the Holling type II functional response: { x ′ ( t ) = x ( t ) [ a ( t ) − b ( t ) x ( t − σ ( t ) ) − h ( t ) x ′ ( t − σ ( t ) ) ] − c ( t ) x ( t ) y ( t ) m y ( t ) + x ( t ) , y ′ ( t ) = y ( t ) [ − d ( t ) + f ( t ) x ( t − τ ( t ) ) m y ( t − τ ( t ) ) + x ( t − τ ( t ) ) ] . An example is presented to illustrate the feasibility of our main results.

Qualitative analysis for a ratio-dependent predator–prey model with disease and diffusion

This paper is purported to study a reaction diffusion system arising from a ratio-dependent predator–prey model with disease. We study the dynamical behavior of the predator–prey system. The conditions for the permanent and existence of steady states and their stability are established. We can obtain the bounds for positive steady state of the corresponding elliptic system. The non-existence results of non-constant positive solutions are derived.

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluctuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratio-dependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

Turing instability for a ratio-dependent predator–prey model with diffusion

Ratio-dependent predator–prey models have been increasingly favored by field ecologists where predator–prey interactions have to be taken into account the process of predation search. In this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction–diffusion model in which predator mortality is neither a constant nor an unbounded function, but it is increasing with the predator abundance. We show that analytically at a certain critical value a diffusion driven (Turing type) instability occurs, i. e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion). We also show that the stationary solution becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises. A numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented. Numerical examples are also included.

Permanence for a delayed discrete ratio-dependent [formula omitted]-species predator–prey system with Holling-type II functional response

In this paper, a delayed discrete ratio-dependent predator–prey model with n − 1 competing preys and one predator and Holling-type II functional response is investigated. A set of easily verifiable sufficient conditions which guarantee the permanence of the system are obtained. Our results show that the delays in the system are harmless for the permanence of the system.

Pattern selection in a ratio-dependent predator–prey model

In this paper, we have presented Turing pattern selection in a ratio-dependentpredator–prey model with zero-flux boundary conditions, for which we have given a generalsurvey of the linear stability analysis and determined the condition of Turing instability,and derived amplitude equations for the excited modes. From the amplitude equations, thestability of patterns towards uniform and inhomogeneous perturbations is determined.Furthermore, we have presented novel numerical evidence of typical Turing patterns, andfound that the model dynamics exhibits complex pattern replication: in the rangeμ1 < μ ≤ μ2, the steady state is the only stable solution of the model; in the rangeμ2 < μ ≤ μ4, on increasing thecontrol parameter μ, the sequence Hπ-hexagons -hexagon–stripe mixture stripes -hexagon–stripe mixture -hexagons is observed; and whenμ > μ4, anH0-hexagon–stripe mixture pattern emerges. This may enrich the pattern formation in adiffusive system.

Positive periodic solutions for the neutral ratio-dependent predator–prey model

By using a continuation theorem based on coincidence degree theory, we obtain some new sufficient conditions for the existence of positive periodic solutions for the neutral ratio-dependent predator–prey model with Holling type II functional response.

Spatial pattern formation of a ratio-dependent predator prey model

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator–prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.

PERSISTENCE IN A RATIO-DEPENDENT PREDATOR-PREY-RESOURCE MODEL WITH STAGE STRUCTURE FOR PREY

This paper deals with a ratio-dependent predator-prey model where the prey population is stage-structured consisting of immature and mature stages and the predator population is influenced by the resource biomass. By means of a transformation of variables, we transform the model into a dynamical system in such a way that there is one-to-one correspondence between the positive values of the original model and the positive values of the transformed model, so that the results which are true for the transformed model are also true for the original model. Dynamical behaviors such as positivity, boundedness, stability, bifurcation and persistence of the model are studied analytically using theory of differential equations. Computer simulations are carried out to substantiate the analytical findings. It is noted that the influence of the resource biomass on the predator population may lead to the extinction of prey population at a lesser value of maturity time in comparison to the absence of the resource biomass.

Qualitative analysis of a stochastic ratio-dependent predator–prey system

A stochastic ratio-dependent predator–prey model is investigated in this paper. By the comparison theorem of stochastic equations and Itô’s formula, we obtain the global existence of a positive unique solution of the ratio-dependent model. Besides, a condition for species to be extinct is given and a persistent condition is established. We also conclude that both the prey population and the ratio-dependent function are stable in time average. In the end, numerical simulations are carried out to confirm our findings.

EXISTENCE OF GLOBAL SOLUTIONS FOR A PREY–PREDATOR MODEL WITH CROSS-DIFFUSION

In this paper, a ratio-dependent prey–predator model with cross-diffusion and homogeneous Neumann boundary condition is studied. Using the energy estimates and the bootstrap arguments, the global existence of solutions for the model is investigated when the space dimension is less than ten.