Study On Predator-prey System With Prey Refuges And Harvesting

In the process of prey population evolution, prey refuges reduce predation as a effectivestrategy.The effect of prey refuges on predator-prey models becomes the hot issue intheoretical ecology and mathematical ecology. The dynamic behavior of a predator-preysystem incorporating refuges was investigated by many scholars. Many important andmeaning conclusions have been obtained: the effect of prey refuges can increase the preypopulation density, the predator population density is increased first and then decreased,increase the stability of the positive equilibrium of the system, that is prey refuge has stability.And ecological systems are generally affected by the human production activities, so it isnecessary to consider human capture for the dynamic behavior of the system, reasonablefishing for the growth of population and the protection of the ecological environment is verymeaningful.Prey usually have two ways to escape from predators, one is through fast running orclimbing; the other is hidden, that is prey come into prey refuges to avoid predators.Therefore, the prey refuges have an effect on the population density of two aspects, on the onehand is to reduce the death rate of prey populations, prey escape into the prey refuges toescape from predators, reducing the number of prey and the death rate of prey populations; onthe other hand, since the number of prey populations that predator is able to catch reduced,leading to predators dying due to lack of food, therefore, the prey refuges played a positiveeffect on prey populations, while played a negative effect on predator populations. The other,because the survival environment of prey refuges is relatively poor, lack of food resources,resulting in reducing the number of new born prey. After a long evolution, making the preypopulations can handle both the positive and the opposite effects coming into prey refuges.The dynamic behavior of predator-prey system with refuges and harvesting in theirspecies are investigated in this paper. Prey refuge and the capacity of independent fishing onthe density of prey and predator is discussed. The effect of prey refuges can increase the preypopulation density, the predator population density is increased first and then decreased,increase the stability of the positive equilibrium of the system, that is prey refuge has stability.Harvesting has an effect on prey population and predator population. If the optimal capacityof independent fishing can be found, it not only stabilize the balance of the ecosystem, butalso meet human’s needs.In Chapter one, the development and meaning of predator-prey system with refuges andharvesting are introduced.In Chapter two, some knowledge related to this paper are introduced. Including the basicconcepts of the population dynamics, for example, equilibrium, Routh-Hurwitz matrix conditions, the conditions of the existence of limit cycles and Pontryagin’s maximal principle,etc.In Chapter three, a Rosenzweig Type predator-prey model is considered. Establish themodel, calculate all the equilibrium points of the model; analysis stability of the equilibriumpoint, according to the Hurwits criterion, using the method of characteristic root, linearized atthe equilibrium, obtained the Jacobi matrix J, when two eigenvalues of J are non zeronegative real part, the equilibrium point of the model is the local asymptotic stable, otherwise,the equilibrium is unstable. The global stability of the positive equilibrium is discussed andobtained the conditions of the global stability of the positive equilibrium and the existence oflimit cycle; according to the Pontryagin maximal principle, the problem of optimal harvestpolicy is solved. Just find the optimal equilibrium point of the model, the optimal fishingintensity can be uniquely determined.In Chapter four, an Ivlev predator-prey system is considered. Build the model, solute allthe equilibriums of the model, analysis the existence and stability of the equilibriums, obtainthe Jacobian by using Routh-Hurwitz, if the two eigenvalues of the Jacobian have nonzeronegative real part, the equilibrium of the model is locally stable; otherwise, the equilibrium isunstable. The global stability of the positive equilibrium is discussed and obtained thecondition of the global stability of the positive equilibrium and the existence of limit cycle;according to the Pontryagin maximal principle, the problem of optimal harvest policy issolved. Only if will we find the optimal equilibrium, the optimal harvesting effort will bedetermined.In Chapter five, by analyzing we find that prey refuge and the harvesting effect mayincrease the stability of the system. If the optimal equilibrium can be found, the optimalcapacity of independent fishing will be ensured. Thereby it can control people’s fishing.Therefore, to meet human’s needs, while on the one hand stabilize the balance of theecosystem, on the other hand strengthen the protection of biological resources, it is of greatsignificance.