Mathematical Models Of Biological Populations

Population ecology is both one important branch of ecology and one branch which mathematics is applied most widely and deep, developing most systematic and adult in ecology heretofore. Linear algebra, differential equation, integral equation, difference equation, functional differential equation, dynamical system, stochastic process, statistical method, or even operator semi-group theory etc are important and staple theory and instrument, mathematics ecological content is studying the mathematical model in population ecology or even still universal ecology with these theory and method. Whereas differential equation model is a sort of all-important model in population ecology, including some well-known important equation thereinto, such as: Malthus equation, Logistic equation and Lotka-Volterra equation, these equations are full important for studying the ecological relationship of population growth.This paper contains four chapters. In chapter 1, the propaedeutics needed in this paper, i.e. theory of stability of the differential equation is introduced briefly. In chapter 2, biotic population 'preliminary model are introduced, i.e., Malthus model, Logistic model of the single population and each other competitive, interdependent mathematical model between two populations. In chapter 3, the generalized Logistic equation is discussed,the functions b and c are assumed to be continuous and positive in R. The following three theorems are obtained and are proved. Theorem 3.1 Let . If u is a positive solution, thenprovided that the limit on the right side exists.Theorem 3.2 If coefficients b and c are T-periodic, then there exists exactly one positive T-periodic solution of (3.1). Theorem 3.3 Let the coefficients b, c be positively bounded. Then equation (3.1) has exactly one positively bounded solution u*(t) on R;and if u is anypositive solution, then u(t) - u* (t)â†' 0 as tâ†'∞.In chapter 4, the predator-prey model of Lotka-Volterra is introduced first,x = x(a-by), y = y(-c + dx), (4.1)a, b, c, d are positive constants.The following theorem and corollary are obtained.Theorem 4.1 For -∞ < α < B , the level sets K_α=F^-1(α) are closed Jordan curves that surround the stationary point (x₀,y₀). all positive solutions (x(t),y(t)) of the Lotka- Volterra equations are periodic; x(t) has its largest and smallest values when y(t) = y₀, and y(t) has its largest and smallest values when x(t) = x₀.Corollary the mean value (x_m,y_m) of a T-periodic solution of equation (4.1), i.e.,is equal to the value of the stationary solution, i.e. x_m = x₀, y_m = y₀.Whereafter the generalized predator-prey models are discussed, the homologous theorem is gotten and is proved.Theorem 4.2 For an autonomous system of the formx|. = W(x,y)h|-(y), y|. = -W(x,y)g|-(x), (4.2)with W > 0. Let the functions g|- ,h|- be continuous and strongly monotone decreasing in [0, ∞) and let each function have a positive zero, say g|-(x₀) = 0, h|- (y₀) = 0. Then (a) The function F(x, y) = G(x) + H(y) withis constant along each solution of the system(4.2). (b) If it is assumed that G(0+) = H(0+) = -∞, then the statements of Theorem 4. are valid.