Analysis Of Stability And Hopf Bifurcation In A Model Of Fishery Resources With Different Cost Functions

In this paper,we mainly study the equilibrium state of the fishery resources.In the construction of ecological civilization environment,analysis of the mathematical economic model with results of the dynamic system provides a general method for the protection of other species.Firstly,this paper introduces the research background and significance and gives domestic and abroad research results in different points of view.Secondly,we bulid different models.The first model: A short-term cost function is a linear function about harvest.We know the changes of market price are negative correlation with the market demand and harvest in a moment.Based on the different benefit goals,considering the bounded rationality fishing establishes a interaction model of fishery resources storage,harvest,and the market price;The second model: Under the condition of long-term we change the linear fishing cost function into nonlinear and keep other conditions being equal.Thirdly,We use the theory of dynamic system to analyze the existence and stability of fixed points of two models;Studying on conditions of Hopf bifurcation in three dimensional fishery resources system about adjustment speed of harvest and market price to obtain corresponding theorems.Under conditions of satisfaction of the theorem,we use the MATLAB programming to draw the time sequence diagram and phase diagram through the numerical simulation for visual analysis.Finally,we obtain the conclusions.The first conclusion: when the cost function is linear,the non-hyperbolic fixed point is semi-stable without effect from the initial point selection.And the hyperbolic fixed point a stable,another is not stable;The second conclusion: when cost function is nonlinear,the non-hyperbolic fixed point is semi-stable and relates to the selection of initial point and coefficient.Hyperbolic fixed point stability also has the corresponding coefficient for strict control;Third conclusion: in the case of giving a part of the fixed parameter in second model,there is a Hopf bifurcation about adjustment speed of harvest market price and bifurcation diagram is obtained by numerical simulation.