Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.     

Qualitative behaviour of a ratio-dependent predator–prey system

This paper deals with the qualitative properties of an autonomous system of differential equations, modeling ratio-dependent predator–prey interactions.This model differs from traditional ratio dependent models essentially in the predator mortality term, the death rate of the predator is not constant but instead increases when there is overcrowding.We incorporate delay(s) into the system. The most important observation is that as the delay(s) is (are) increased the originally asymptotic stable interior equilibrium loses its stability. Furthermore at a certain critical value a Hopf bifurcation takes place: small amplitude periodic solutions arise.     

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.     

Modelling a predator–prey system with infected prey in polluted environment

A predator–prey model with logistic growth in prey is modified by introducing an SIS parasite infection in the prey. We have studied the combined effect of environmental toxicant and disease on prey–predator system. It is assumed in this paper that the environmental toxicant affects both prey and predator population and the infected prey is assumed to be more vulnerable to the toxicant and predation compared to the sound prey individuals. Thresholds are identified which determine when system persists and disease remains endemic.     

Turing pattern formation in a predator–prey–mutualist system

In this paper, we develop a theoretical framework about spatial patterns in a three-species predator–prey–mutualist system with cross-diffusion. We concentrate on three aspects of Turing pattern formation: (1) what conditions enable the occurrence of Turing patterns? (2) what are the underlying mechanisms? (3) what are the corresponding configurations? For the first two questions, by use of the stability analysis for the positive uniform solution and the Leray–Schauder degree theory, we prove that under some conditions, the system admits at least a nonhomogeneous stationary solution. For the third question, we carry out numerical simulations for a Turing pattern, and we show that the configurations of Turing pattern are stable spotted patterns, which resemble a real ecosystem.     

Bifurcation analysis in a discrete differential-algebraic predator–prey system

The discrete-time predator–prey biological economic system obtained by Euler method is investigated. Some conditions for the system to undergo flip bifurcation and Neimark–Sacker bifurcation are derived by using new normal form of differential-algebraic system, center mainfold theorem and bifurcation theory. Numerical simulations are given to show the effectiveness of our results and also to exhibit period-doubling bifurcation in orbits of period 2, 4, 8 and chaotic sets. The results obtained here reveal far richer dynamics in discrete differential-algebraic biological economic system. The contents are interesting in mathematics and biology.     

Dynamics of a predator–prey system with inducible defense and disease in the prey

Establishing and researching a population dynamical model based on the differential equation is of great significance. In this paper, a predator–prey system with inducible defense and disease in the prey is built from biological evolution and Eco-epidemiology. The effect of disease on population stability in the predator–prey system with inducible defense is studied. Firstly, we verify the positivity and uniform boundedness of the solutions of the system. Then the existence and stability of the equilibria are studied. There are no more than nine equilibrium points in the system. We use a sophisticated parameter transformation to study the properties of the coexistence equilibrium points of the system. A sufficient condition is established for the existence of Hopf bifurcation. Numerical simulations are performed to make analytical studies more complete.     

Dynamics in a diffusive predator–prey system with a constant prey refuge and delay

In this paper, a diffusive predator–prey system with a constant prey refuge and time delay subject to Neumann boundary condition is considered. Local stability and Turing instability of the positive equilibrium are studied. The effect of time delay on the model is also obtained, including locally asymptotical stability and existence of Hopf bifurcation at the positive equilibrium. And the properties of Hopf bifurcation are determined by center manifold theorem and normal form theorem of partial functional differential equations. Some numerical simulations are carried out.     

Bifurcation and chaotic behavior of a discrete-time predator–prey system

The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant ${\mathbb{R}}_{+}^2$. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of ${\mathbb{R}}_{+}^2$ by using a center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as orbits of period 7, 14, 21, 63, 70, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, quasi-periodic orbits and chaotic sets. These results show far richer dynamics of the discrete model compared with the continuous model. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

The effect of refuge and immigration in a predator–prey system in the presence of a competitor for the prey

This paper considers the effect of immigration and refuge on the dynamics of a three species system in which one predator feeds on one of two competing species. Immigration is assumed only for the species which is not attacked by the predator.The main results address the stability of the system. Namely, it is shown that increasing the number of refuges stabilizes the system, whereas the opposite holds true by increasing the immigration rate. Also, one result about the persistence of the system and one concerning the global stability of the coexistence equilibrium are presented. Some numerical simulations illustrate the obtained results.     

Impact of cannibalism on dynamics of a structured predator–prey system

Cannibalism, as a behavioral trait, is prevalent in many species. To have better understanding of their dynamics, we investigate a structured predator-prey system with predator cannibalism, where the prey population follows the logistic growth in the absence of the predator. We study the effects of the cannibalism attack rate and the corresponding benefit rate of cannibals on the model dynamics. Complex phenomena, including the bistability, the existences of two positive equilibria and stable/unstable periodic solutions, are found. We define quantities with clear biological meanings, and establish conditions determining the local and global dynamics of the model based on these quantities. Our results show that, under certain conditions, the final states of the populations depend on not only the related model parameters but also the initial conditions of the solutions.     

Uniqueness of periodic solutions of a nonautonomous density-dependent predator–prey system

In this present paper, we investigate the uniqueness of periodic solutions of a nonautonomous density-dependent and ratio-dependent predator–prey system, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. We start with a sufficient condition for the permanence of the system and then construct a weaker sufficient condition by introducing a specific set, denoted as Γ. Based on this Γ and the Brouwer fixed-point theorem, we obtain the existence condition of positive periodic solutions. Moreover, since the uniqueness of positive periodic solutions can be ensured by global attractiveness, we alternatively introduce a sufficient condition for global attractiveness. Similarly, we also provide a sufficient condition for the uniqueness of non-negative periodic solutions.     

Dynamic analysis of a fractional order prey–predator interaction with harvesting

In recent years, prey–predator models appearing in various fields of mathematical biology have been proposed and studied extensively due to their universal existence and importance. In this paper, we introduce a fractional-order prey–predator model and deals with the mathematical behaviors of the model. The dynamical behavior of the system is investigated from the point of view of local stability. We also carry out a detailed analysis on the stability of equilibrium. Numerical simulations are presented to illustrate the results.     

Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey

This work deals with the analysis of a predator–prey model derived from the Leslie–Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered.