The stability and bifurcation analysis for a predator–prey system with the nonlinear Michaelis–Menten type predator harvesting are taken into account. The existence and stability of possible equilibria are investigated. Specially, the stability of some positive equilibria is determined by using numerical simulation method due to the fact that the expressions of determinant and trace of the Jacobian matrix at these equilibria are very complex. The rigorous mathematical proofs of the existence of saddle–node bifurcation and transcritical bifurcation are derived with the help of Sotomayor’s theorem. Furthermore, in order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Choosing two parameters of the system as bifurcation parameters, we prove that the system exhibits Bogdanov–Takens bifurcation of codimension 2 by calculating a universal unfolding near the cusp. Numerical simulations are carried out to demonstrate the validity of theoretical results. Our research will be useful for understanding the dynamic complexity of ecosystems or physical systems when there is the nonlinear Michaelis–Menten type harvesting effect on predator population. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant-yield harvesting and constant-effort harvesting. It can be thought as a supplement to existing literature on the dynamics of this system, since there is little literature involved in nonlinear type harvesting for the system up to now.

Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting

Turing instability and pattern formation in a super cross-diffusion predator-prey system with Michaelis-Menten type predator harvesting are investigated. Stability of equilibrium points is first explored with or without super cross-diffusion. It is found that cross-diffusion could induce instability of equilibria. To further derive the conditions of Turing instability, the linear stability analysis is carried out. From theoretical analysis, note that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes by means of weakly nonlinear theory. Dynamical analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, the theoretical results are illustrated via numerical simulations.

Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting

Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semi-analytical and numerical solutions of the governing equations.

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The diffusive Lotka-Volterra predator-prey system with delay.

Mobile wireless sensor networks (MWSNs) have become an area of intense research activity due to technical advances in sensors, wireless communications and networking, and signal processing. Many applications, including environment monitoring, battlefield surveillance, and urban search and rescue especially in hazardous situations, are envisaged. However, MWSNs may be vulnerable to malicious interference because of the large-scale characteristics. When a contaminated node communications with its neighbors, multiple copies of the malware are transmitted to its neighbors, which may destroy, block regular communications, or even damage the integrity of regular data packets. Modeling spatial distribution of malware propagation over time is the first step to predict the trend of malware propagation in MWSNs. We propose a novel wireless malware propagation model with the discrete time delay based on reaction-diffusion equations in mobile wireless sensor networks, and study its dynamic behaviors. By analyzing the stability and Hopf bifurcation of the equilibrium of our model, we search for the sufficient conditions, which leads to the malware propagation disappears or continues. Furthermore, we demonstrate that oscillations in this model occur through the destabilization of the stationary solution at a Hopf bifurcation point. And formulas for determining the stability of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Finally, we conduct extensive simulations on large-scale MWSNs to evaluate the proposed model. Numerical evidence shows that the spatial-temporal dynamic characteristics of malware propagation in MWSNs are closely related to the packet transmission rate, the communication rang and the mobile behavior of nodes.

Stability and bifurcation analysis in a delayed reaction-diffusion malware propagation model

In this paper, a singular phytoplankton–zooplankton model with fuzzy parameters, refuge, fishery protection and harvesting is studied by regarding the imprecise biological parameters as one form of triangular fuzzy number. By using the utility function method, the transformed differential equation is reduced to a differential equation. In absence of economic profit, the existence of positive equilibria, local stability and global stability for the interior equilibrium of the present differential system are discussed, the conditions of Hopf bifurcation occurring at the positive equilibrium are given, and the corresponding nonlinear feedback controller is designed to eliminate Hopf bifurcation. In presence of economic profit, the corresponding differential algebraic model is proposed and some conditions of the existence of the interior equilibrium are derived. By regarding economic profit as bifurcation parameter, singular induced bifurcation occurring is investigated, and a linear controller is designed to remove such bifurcation. Furthermore, Pontryagin’s maximum principle has been used to obtain optimal harvesting policy to maximize the benefit as well as conservation of the ecosystem. Finally, some numerical simulations are given to demonstrate our theoretical results after each part. Results show that imprecise parameters not only affect the interior equilibrium and the bionomic equilibrium of system, but also affect the critical value of bifurcation and branch range.

Hongyong Zhao

Papers

Bifurcation and optimal harvesting of a diffusive predator–prey system with delays and interval biological parameters ☆