A General "Food-limited' Species Model With Random Perturbation

The study about mathematical biology become more and more important in the 21th century. The intersectant domain will be researched mainly between mathematical biology and other subjects. Compared with the determinate mathematical biology model, species ecology systems are often subject to environmental noise, it is important to discover whether the presence of a such noise affects these results that we have obtained. Moreover the estimating of the parameters by applying statistical methods will become an important disscussion with applying stochastic differential equation extensively in the mathematical biology .We conside a randomed general "food-limited " species modelwhere θ is a positive odd number^[4,11] and B(t)(t > 0) is the 1-dimensional standard Brownian motion with 5(0) = 0. Initial value N(0) = N0 and N0 is a positive number. By the Ito interpretation, Eq(1.2) is equivalent toThe existence,uniqueness, global stability of positive solutions and the estimate of parameters of Eq. (1.3) are given by using both stochastic differential equation theories and numerical simulation .The paper first proves that the solution of Eq. (1.3) is positive and global. Thus we can make the change of logarithm variable and the existence of positive solutions is proved indirectly . Moreover the coefficients of Eq. (1.3) statisfy the local Lipschitz condition. Therefore Eq.(1.3) has a unique continuous global positive solution with any initial value N0 > 0. The global stability in the (9th moment and the (6 + l)th moment of Eq. (1.3) satisfying initial value 0 < N0 < K are given with help of Lyapunov's second method. We alse carry numerical simulations for the concrete model, which strongly support our theoretical studies. At last we obtain the Maximum Likelihood Estimate (MLE) of the parameters of Eq. (1.3). Simulation results show that MLE fits well.