Research On A Stochastic Population Model With Age-structure

Population ecology is a sub-field of the ecology, for the study the change of populationsize over time, space, or other factors change. There are significant differences in thebiological characters of all ages in practice, so the birth rate, mortality is completely different.In addition, the biological characteristics of the population can not only be affected in thesurvival process by the external random factors, but also affected by some special naturalphenomena, such as earthquakes, fires, snowstorms and others. Therefore, consideringage-structured stochastic population models with Markovian Switching is a very importanttheoretical significance and practical value.The convergence of the system by constructing the semi-implicit Euler discrete systemsof age-structured stochastic population models with Markovian Switching was studied, andits numerical verification was showed, finally we obtained the following main conclusions:1. The convergence of the discrete system of this model was discussed by using Itoformula, Gronwall inequality, Burkholder-Davis-Gundy inequality and some basic tools withbasic knowledge of mathematics. The numerical method converged the real solution of thepopulation model in mean square, so it was very reliable to study numerical simulation ofsuch a model by the use of semi-implicit the Euler numerical methods.2. According to the characters of the discrete system, combined with the MATLABsoftware, we achieved the semi-implicit Euler method of age-structured stochastic populationmodels with Markovian Switching, showed the flow chart of this numerical process, anddiscussed the differences of different numerical results with different segmentations:a. Without age, population density showed dramatic fluctuations in trends over time, thistrend was due to random factors outside and gradually decreased with the increase in time.The beginning of the dramatic fluctuations in population density was that at the beginning,individuals in the population, in order to better survival and development, competed againstother individuals in the population to natural resources, prey and other factors, therebyaffecting the population changes obviously. The following decrease was due to the number ofpopulation growth to close to the maximum carrying capacity of the environment, and the constraints of the natural resources, prey and other factors.b. The change trend of S-shaped in population density gradually decreased in the case ofnot considering time. When age a [0,0.1], due to the higher number of adult groups withinpopulation, the birth rate was slightly higher than the mortality rate, so the population showeda growth trend; when age increases, the number of older age groups within the populationstarted to grow, which led to the birth rate gradually decreases and the death rate had agrowth, so population density started to decrease, finally reaching the dynamic equilibrium ofpopulation density.3. This paper compared and discussed the numerical results of different numericalmethods by different, that is semi-implicit the Euler numerical methods and implicitnumerical methods, overall the results showed that the numerical results of the two numericalmethods were similar.