Study On Dynamic Behavior And Optimal Control Of Population Excited By Color Noise

The exploitation and utilization of population resources is a classic problem in academic circles.For the study of this problem,human beings should not only consider the short-term economic benefits,but also pay more attention to the exploitation and sustainable utilization of biological resources.As for biological evolution,population is a basic unit and an important part of the ecosystem.The growth of population is an important index reflecting the ecological balance of population.By introducing the classical population model,the dynamic behavior and optimal control of the population induced by color noise are studied,which is of great significance to fully exploit and protect the population resources.Study of dynamic behavior and optimal control of population system under noise excitation provides a new idea and method for maintaining ecological balance,protecting biodiversity,regulating the relationship between species and effectively exploiting and utilizing biological resources.The effects of color noise and Allee effect on population dynamic behavior and optimal control are systematically analyzed in this paper.The main work and conclusions are as follows:1.Stochastic dynamics behavior and optimal control of single population systems with Gaussian color noise are studied.Firstly,the Fokker-Planck equation is obtained by the unified color noise approximation(UCNA),and the steady-state density function and the mean first crossing time of the population are further obtained.Then,we simulate and discuss the effects of different variables on stationary probability distribution(SPD)and mean first crossing time,and obtain the resonance activation phenomena of multiplicative noise intensity and population growth rate.Finally,on the basis of sustainable income and maximum retained profit,the expected maximum value and corresponding variance of optimal capture control are obtained,and the theoretical results are analyzed and verified by representative examples.This study shows that the random disturbance of external environment will have a negative impact on the exploitation and utilization of population resources,and the increase of multiplicative and additive noise intensity is not conducive to the growth and reproduction of population,and increases the possibility of population extinction.2.The stochastic dynamic behavior and optimal control of a single population system with Allee effect under non-Gaussian color noise are studied.Firstly,the non-Gaussian color noise is simplified and approximated to gaussian color noise,and then the FPK equation is obtained by unified color noise approximation.Then the influence of different variables on steady-state probability distribution and mean first crossing time is simulated and discussed.Finally,the maximum expectation and corresponding variance of population optimal control are obtained.In this paper,it can be seen that under strong Allee effect and weak Allee effect,the increase of non-Gaussian noise intensity can lead to phase transition and increase the probability of population extinction.Under strong Allee effect,small Allee threshold is beneficial to population survival.The increase of correlation time of multiplicative non-Gaussian color noise will prolong the transition time from extinction state to survival state,which is not conducive to the survival and development of population.