Bifurcation analysis of a diffusive predator–prey model with ratio-dependent Holling type III functional response

In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction–diffusion predator–prey system with mutual interference described by the Crowley–Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction–diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction–diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

Small perturbations in the type of boundary conditions for an elliptic operator

In this paper, a type of reaction–diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered. By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator, the characteristic equation of this system is obtained. Then, by selecting time delay and self-feedback strength as the bifurcating parameters respectively, the dynamic behaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary. Furthermore, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation. Finally, two simulation examples are given to verify the theory.

Stability and Hopf bifurcation for a three-species food chain model with time delay and spatial diffusion

In this paper, a three-species Lotka-Volterra food-chain model with spatial diffusion and time delays is investigated. We first analyze the local stability of the steady states and the existence of Hopf bifurcation to this system under homogeneous Neumann boundary conditions. We consider the effects of impulses on the dynamics of the above food-chain model without spatial diffusion. Numerical simulations show that the system with constant periodic impulsive perturbations admits rich complex dynamics. Key words: Hopf bifurcation, food chain, reaction diffusion, delay, stability, chaos.

In this paper, a predator-prey model with stage structure for the predator and a spatio-temporal delay describing the gestation period of the predator under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states is established. Sufficient conditions are derived for the global attractiveness of the positive steady state and the global stability of the semi-trivial steady state of the proposed problem by using the method of upper-lower solutions and its associated monotone iteration scheme. Numerical simulations are carried out to illustrate the main results.

Numerical modelling of an SIR epidemic model with diffusion

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

Abstract. This paper is concerned with the positive steady states of adiffusive Holling type II predator-prey system, in which two predatorsand one prey are involved. Under homogeneous Neumann boundary con-ditions, the local and global asymptotic stability of the spatially homoge-neous positive steady state are discussed. Moreover, the large diffusion ofpredator is considered by proving the nonexistence of non-constant posi-tive steady states, which gives some descriptions of the effect of diffusionon the pattern formation. 1. IntroductionDue to the universal existence of energy transformation, predator-prey sys-tem is very important in describing the population evolution, and it is oneof the dominant themes in both ecology and mathematical ecology. Becauseof the differences in capturing food and consuming energy, a major trend intheoretical work on predator-prey dynamics has been launched so as to derivemore realistic models and functional responses, for example, Lotka-Volterratype [22, 31], Holling type [14], Beddington-DeAngelis type [3, 5] and so on.These models often involve only one predator species and one prey species. Inreality, it is very common that one species is captured by several species in foodchains. In particular, Hixon and Jones [13] found that the density-dependentmortality in demersal marine fishes is often caused by interaction of predationand competition. In a very recent paper, to study how the nonlinear mortalityrate determines the dynamics of such competition models, Ruan et al. [29]proposed the following two predators and one prey model with Holling II type

Dynamics in a diffusive plant–herbivore model with toxin-determined functional response

Non-constant steady states and Hopf bifurcation of a species interaction model

A Hölling’s type II prey–predator model with stage structure and nonlocal delay

In this article, we propose and analyze an infectious disease model with reinfection and investigate disease dynamics by incorporating saturated treatment and information effect. In the model, we consider the case where an individual’s immunity deteriorates and they become infected again after recovering. According to our findings, multiple steady states and backward bifurcation may occur as a result of treatment saturation. Further, if treatment is available for all, the disease will be eradicated provided {mathcal {R}}_0<1; however, because limited medical resources caused saturation in treatment, the disease may persist even if {mathcal {R}}_0<1. The global stability of the unique endemic steady state is established using a geometric approach. We also establish certain conditions on the transmission rate for the occurrence of periodic oscillations in the model system. Among nonlinear dynamics, we show supercritical Hopf bifurcation, bi-stability, backward Hopf bifurcation, and double Hopf bifurcation. To illustrate and validate our theoretical results, we present numerical examples. We found that when disease information coverage is high, infection cases fall considerably, and the disease persists when the reinfection rate is high. We then extend our model by incorporating two time-dependent controls, namely inhibitory interventions and treatment. Using Pontryagin’s maximum principle, we prove the existence of optimal control paths and find the optimal pair of controls. According to our numerical simulations, the second control is less effective than the first. Furthermore, while implementing a single intervention at a time may be effective, combining both interventions is most effective in reducing disease burden and cost.

A delayed predator–prey diffusion system with homogeneous Neumann boundary condition is considered. In order to study the impact of the time delay on the stability of the model, the delay τ is taken as the bifurcation parameter, the results show that when the time delay across some critical values, the Hopf bifurcations may occur. In particular, by using the normal form theory and the center manifold reduction for partial functional differential equations, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution have been established. The effect of the diffusion on the bifurcated periodic solution is also considered. A numerical example is given to support the main result.