The Global Attractivity Of Several Types Of Delay Logistic Equations

In the Chapter 1:the delay logistic equations are introduced in the history of research, application of these equation.In the Chapter 2, we mainly introduce the global attractivity of several delay Logistic equations. We show the global attractivity conditions with respect to the concrete Logistic equations.Firstly, we show the global attractivity in nonautonomous logistic equation with delay. It is well known that most of the delay Logistic equaitons are nonautonomous, for example, the following linear equations x′(t)= x(t)[r(t)-a(t)x(t)-b(t)x(t-τ)], (1) where r(t), a(t), b(t) are continuous, T-periodic and, m[r(t)]> 0, a(t)> 0, b(t)≥0. (2)We consider the global attractivity of the equation below x'(t)= (1+x(t))(-a(t)x(t)-β(t)x(t-τ)). (3) where a(t),β(t) are continuous, T-periodic and, a(t)> 0,β(t)≥0.Theorem 1 [7, Theorem3.1] Suppose coefficients a(t) andβ(t) of Eq. (3) verify the follow-ing two conditions: (Hi) There exists a continuous, periodic function y(t), such that m[y]> 0(H2) For any t0> 0,there exists k> 0 (depending y(t), and t0 such that m[y]> 0, Then the zero solution is globally attractive with respect to solution x(t) of (3), x(t>-1).Theorem 2 [7, Theorem4.3] Let x(t) be a positive perioditic solution of (1), Assume that assumptions (Hi), (H2) of Theorem 1 hold for a(t)=a(t)x(t),β(t)=b(t)x(t-τ). ThenSecondly, we introduce permanence and global attracitivity of a delayed periodic logis-tic equation. The following equation is shown as an example We only consider the initial conditions where a∈C(R, R) and b, c,σ,τ∈C(R, [0,∞]) areω-periodic function with p≥q are positive constants.For system (4) we always assume that(H1) a(t), b(t), c(t) are continuous,positiveω-periodic function define on [0,∞),p≤q are positive constants. (H2)σ(t),τ(t) are nonnegative continuously differentiableω-periodic function on [0,∞) whereσ(t)=dσ(t)/dt,τ(t)=dτ(t)/dt.Theorem 3 [8, Theorem 3.1] In addition to (H1), (H2)assume further that (H3) and (H4), where M0 are defined as follow. Then for any two positive solution N1 (t) and N2(t) of system (4), one hasThirdly, we introduce global attractivity for a delay logistic equation with instantaneous terms. The following equation with be introduced as an example, with the corresponding initial value condition where c≥0, r(t)∈C([0,∞), (0,∞)), F:[0,∞)×BC→R and for anyψ∈C([g(0),∞),R) withψt∈BC, the function t→F(t,ψt) is continuous on [0,∞), and satisfies that Theorem 4 [28, Theorem 3.1] Assume that c> 1 an (8) and (9) hold. Then every solution of (7) with (6) tends zero ast→∞.Theorem 5 [28, Theorem 3.2] Assume that c= 1 and (8), (9) and (10) hold. Then every solution of (7) with (6) tends zero ast→∞.Theorem 6 [28, Theorem 3.3] Assume that 0< c< 1 and (8), (9) and (11) hold. Then every solution of (7) with (6) tends zero as t→∞.Forthly, we show the global attractivity and positive almost periodic solution for delay logistic differential equation. The following equation will be discussed as an example,Theorem 7 [37, Theorem 6] Assume that a(t), b(t)and r(t) are almost periodic, a*> 0, and b*＞x1/x2. Then (12) has a unique positive almost periodic solution p(t) with mod(p(t))(?) mod (a(t), b(t), r(t)) and p(t) is globally attractive.Theorem 8 [37, Theorem 7] Assume that a(t), b(t) and r(t) are almost periodic functions, Then (12) has a unique positive solution p(t) with mod (p(t)) (?) mod (a(t), b(t), r(t)) and p(t) is globally attractive.At last, we introduce the global attractivity for a logistic equation with piecewise con- stant argument. The following equation as an example, is introduced with the initial condition x(0)= x0> 0, where [·] denotes the greatest integer function, r:[0,∞)→[0,∞) is a continuous function and, K is a positive constant. For any nonnegative integer t, x'(t) means the right-hand derivative of x(t). The equation (13) has been considered as a semi-discretization of (14) the delay logistic equationTheorem 9 [42, Theorem A] Assume that and Then the positive state x(t)= K of (13) is globally attractive. (That is, every solution y(t) of (13) converges to K as t→∞).Theorem 10 [40, Theorem 1.1] Assume that and Then the positive state x(t)= K of (13) is globally attractive.In the Chapter 3, we mainly give the analysis of difference of the global attractivity of several delay Logistic equations, including the application, the methodology, and so on.