A prey-predator model with harvesting for fishery resource with reserve area

Considering that over exploitation would result in the extinction of the population, we propose and investigate a Holling II functional response prey-predator model with harvesting for fishery resource in a two-patch environment: a free fishing zone (patch 1) and a reserve zone (patch 2) where fishing is strictly prohibited. First, the presence of harvesting can impact the existence of equilibria. Further, stability criteria of the model is analyzed both from local and global point of view. Our results indicate that so long as the prey population in the reserved zone does not extinct, the both prey always exist, that is marine reserves should ensure the sustainability of system. Thus, marine reserves not only protect species inside the reserve area but they can also increase fish abundance in adjacent areas. Next, the existence of bionomic equilibrium and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle. It is established that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.     

一类带有交叉扩散的捕食模型的共存问题

研究带有齐次Dirichlet边界条件的捕食-食饵模型,得到了平凡解存在的条件,并给出半平凡解存在的充分条件以及解的先验估计,最后利用Shauder不动点定理,得到问题至少有一个正解存在的充分条件.该结果说明只要捕获率足够小,物种的交叉扩散相对弱,问题就至少存在一个正解.     

Global dynamics of a two-patch SIS model with infection during transport

A two-patch SIS model is formulated and studied. The standard incidence rate and mass-action incidence rate are used within each patch and during transport, respectively. The basic reproduction number is calculated and the global dynamics is investigated. The simulation results show the influence of travel rates, the different dynamics by using standard incidence rate and mass-action incidence rate. The importance of border screening is also explored by numerical simulation.     

Stationary patterns of a ratio-dependent prey-predator model with cross-diffusion

This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross-diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.     

Non-constant stationary solutions to a prey-predator model with diffusion

In this paper, a nonlinear predator reproduction and prey competition model with diffusion is discussed. Some existence and non-existence results concerning non-constant positive steady-states are presented using topological degree argument and the energy method, respectively. The first author was supported by the National Natural Science Foundation of China (No. 10771032) and the Natural Science Foundation of Jiangsu province BK2006088, and the second author was supported by National Natural Science Foundation of China (No. 10601011).     

Mathematical analysis of a two-patch model of tuberculosis disease with staged progression

The spread of tuberculosis is studied through a two-patch epidemiological system SE₁ ? E_nI which incorporates migrations from one patch to another just by susceptible individuals. Our model is consider with bilinear incidence and migration between two patches, where infected and infectious individuals cannot migrate from one patch to another, due to medical reasons. The existence and uniqueness of the associated endemic equilibria are discussed. Quadratic forms and Lyapunov functions are used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium (DFE) is globally asymptotically stable, and when it is greater than one there exists in each case a unique endemic equilibrium (boundary equilibria and endemic equilibrium) which is globally asymptotically stable. Numerical simulation results are provided to illustrate the theoretical results.     

Analysis of a prey-predator model with disease in prey

In this paper, a system of reaction-diffusion equations arising in eco-epidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the predator is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.     

A minimal model of pattern formation in a prey-predator system

The spatio-temporal dynamics of a prey-predator community is described by two reaction-diffusion equations. It is shown that for a class of initial conditions the spatio-temporal system dynamics resembles a “phase transition” between a regular and an irregular phase, separated by a moving boundary. A simple approach to specify spatio-temporal chaos is proposed.     

Stochastic analysis of prey-predator model with stage structure for prey

The present paper deals with the effect of environmental fluctuation on a prey-predator model with stage structure for prey population. We have studied the stochastic behaviour of the model system around coexisting equilibrium point. Stochastic stability condition in mean square sense is obtained for the stage-structured model with help of a suitable Lyapunov function. Numerical simulations are carried out to substantiate the analytical findings. The main outcomes of mathematical findings are mentioned in conclusion section.     

On a new model of two-patch predator-prey system with migration of both species

In this paper we explore the dynamics of predator-prey in a two patch system. The two patches of the system are coupled with both the migration of the predator and the prey. The purpose of this exploration is to find upper and lower bounds for the populations and get an insight on the different possibilities with the three types of Holling functional responses. Also we discuss the stability and instability of the equilibrium solutions found in earlier papers. Numerical simulations are provided to graphically demonstrate the population dynamics of the system.     

Population model with diffusion and supplementary forest resource in a two-patch habitat

A mathematical model is proposed to study the role of supplementary self-renewable resource on species population in a two-patch habitat. It is assumed that the density of forest resource biomass is governed by the logistic equation in both the regions but with the different intrinsic growth rate but the same carrying capacity in the entire habitat. It is further assumed that the densities of species population is also governed by the generalized logistic equations in both the regions but with different growth rates and carrying capacities. It is shown that the steady state solutions are positive, monotonic and continuous under both reservoir and no-flux boundary conditions. The linear and non-linear asymptotic stability conditions of non-uniform steady state are compared with the case of the model with and without diffusion in a homogeneous habitat.     

Qualitative analysis on a diffusive prey-predator model with ratio-dependent functional response

In this paper, we investigate a prey-predator model with diffusion and ratio-dependent functional response subject to the homogeneous Neumann boundary condition. Our main focuses are on the global behavior of the reaction-diffusion system and its corresponding steady-state problem. We first apply various Lyapunov functions to discuss the global stability of the unique positive constant steady-state. Then, for the steady-state system, we establish some a priori upper and lower estimates for positive steady-states, and derive several results for non-existence of positive non-constant steadystates if the diffusion rates are large or small. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10801090, 10726016, 10771032) and the Scientific Innovation Team Project of Hubei Provincial Department of Education (Grant No. T200809)     

一类带非单调转化率的捕食-食饵模型的全局分歧

主要研究了一类带非单调转化率的捕食-食饵模型,分别以生长率a和b为分歧参数,运用度理论和分歧理论讨论了这类模型在齐次第一边界条件下全局分歧结构．     

Dynamics of a stochastic three species prey-predator model with intraguild predation

Intraguild predation is ubiquitous in many ecological communities. This paper is concerned with a stochastic three species prey-predator model with intraguild predation. The model involves a prey, an intermediate predator which preys on only prey and an omnivorous top predator which preys on both prey and intermediate predator. First, we show the existence of a unique positive global solution of the model. Then we mainly establish the sufficient conditions for the extinction and persistence in the mean of each population. Moreover, we show that the model is stable in distribution. Finally, some numerical simulations are given to illustrate the main results.     

Dynamics of a delay-diffusion prey-predator model with disease in the prey

A mathematical model dealing with a prey-predator system with disease in the prey is considered. The functional response of the predator is governed by a Hoilling type-II function. Mathematical analysis of the model regarding stability and persistence has been performed. The effect of delay and diffusion on the above system is studied. The role of diffusivity on stability and persistence criteria of the system has also been discussed.     

Positive solutions of a prey-predator model with predator saturation and competition

In this paper, we study the existence, multiplicity, bifurcation and stability of positive solutions to a prey-predator model with predator saturation and competition

     

Existence of positive solutions for a prey-predator model with refuge and diffusion

We are concerned with a prey-predator reaction-diffusion model with monotonic functional response and specific refuge size. We discuss both existence and nonexistence of positive solutions of the model by using fixed point index theory and bifurcation theory as the main argument tools. We also discuss bifurcation solutions of which stability is established by using spectrum analysis methods. Moreover, we analyze the effects of refuge size on the dynamics of the model.     

A two-patch model for the optimal management of a fishing resource considering a marine protected area

In this work, we propose and analyze a model related with the management optimization of a renewable resource in aquatic environment composed of two different patches. Spatial distribution of each subpopulation is assumed: one is developed in a marine protected area (MPA) or a marine reserve and the other is located in a zone where fishing with open access may be effected.It is generally assumed that there may be migration between both areas, but in this work we will consider that the flux goes.When a fishing ban in the protected area is established it becomes a marine reserve, which can also be assumed as a refuge for the captured species. In this case, the marine reserve is the source and the exploitation area is a sink.The behavior of the renewable resource is modeled by a deterministic continuous time system. To establish the optimal harvesting policy, we will maximize the present value J of a continuous time stream of revenues, given by a cost functional indicating the net economic revenue to the fishermen, the perceived rent. Using Pontragyn’s Maximum Principle we will obtain the Hamiltonian function to determine the optimal policies.