Stability Analysis Of A Single-species Diffusion Model With Memory Effect

More and more studies have shown that: in real life,the biological population model whose coefficient depends on time delay is closer to the entire life course of organisms.Under the premise of continuous changes in the number of biological populations,in order to fully understand the changes and stability of the population,this paper considers factors such as population diffusion and maturity,and then a population diffusion model including memory delay and maturity delay and whose coefficients depend on the delay is established to analyze the stability of the single-species diffusion model with memory effect.First of all,considering the importance of the characteristic equation to the biological population model,we first solve the characteristic equation to prepare for the follow-up research.After simplifying and sorting,the positive equilibrium point with practical significance is obtained,and then the equation is linearized at the equilibrium point,and then the characteristic equation is obtained.After analysis,the stable conditions for the steady-state solution are obtained.On the basis that the characteristic equation satisfies some conditions,we carry out the following research on the feasible region and branch curve.Secondly,since we want to study the branch curve in the feasible region,it is necessary to clearly discuss the feasible region.The characteristic equation is analyzed using geometric methods to finally determine the feasible region,and then the curve C in the feasible region is obtained by numerical simulation,which provides the imaginary part of the characteristic value and the range of the mature delay for the subsequent drawing of the branch curve,and transforms the parameters to obtain the corresponding feasible region,analyze its rules one by one,and then conduct research on branch curves in feasible regions.Finally,considering the maturity delay and memory delay at the same time,within the permitted feasible region,select the appropriate coordinate nodes to obtain the branch curve,and then analyze the shape type of the branch curve;calculate the direction of the branch curve.When the time delay passes through the branch curve in the direction of the branch curve,discuss the direction of the eigenvalue crossing the imaginary axis,analyze the change law of the branch curve,and find that the steady-state solution is "infinite-dimensional" unstable.On the basis of the aforementioned research,the solution of the equation of this model is simulated,and it is verified that the solution of the equation eventually blows up due to the unstable "infinite dimension" of the steady-state solution of the model.