Permanence Of The Nonautonomous Species Dynamics Model With Third Type Holling Function Response

The permanence, existence and globally asymptotical stability of periodic solutions or almost periodic solutions of Lotka-Volterra systems are important aspects of mathematical ecology research. In this paper, we study a nonautonomous Lotka-Volterra model with Holling typeⅢfunctional response and diffusion, a three-species mixed system with HollingⅢfunctional response and stage-structure for prey, and a nonautonomous competition delay system with HollingⅢfunctional response and diffusion for predator. A set of sufficient conditions are obtained for the existence, global asymptotic stability of periodic solutions or positive almost periodic solutions for these three models by using qualitative theory of differential equations, extremum principle, fixed point theorem and V function method, Lyapunov function method etc.The whole paper consists of four chapters.In Chapter 1, we introduce the developing history of the population ecology.In Chapter 2, we consider the permanence, existence and global asymptotic stability of periodic solutions of the following nonautonomous food chain Lotka-Volterra model with HollingⅢFunction response and diffusion for prey. The main conclusions are given by the following theorems.Theorem 2.1 Assume that system (2.1) satisfies the following conditions (H1)～(H4) { }( ) * *( ) max 1 , 2 : 1, 1, 2,ui tixi ti = e≤M M = M i= (H1) where 2()()()4()()max11200[0,]*atatatathtMjjjjjjt=∈ω++ j=1,2; iutxi tieMii( )= ()< i =3,4, (H2) where ()()max22201[ 0,]ataatMjjjjjit=∈ω?+????αβ?????? j =3,4; { }1*2*1x( )e( )minm,mmuiiiτi=τ≥= , i =1,2, (H3) where ,2()()()4()()min1131111121010[0,]*1 atatatathtMmt=∈ω++???? ?????αβ????????m 2 *= ????aa2201????; ( )( )ui ixiτi = eτ≥mi , i = 3 , 4, (H4) Then system (2.1) is permanent.Theorem 2.2 Assume that conditions (H1)～(H4) hold. Then there is at least one strict positiveω-periodic solution.Theorem 2.3 Assume conditions (H1)～(H4) in periodic system (2.1) hold and the following condition is satisfiedThen system (2.1) has a unique strict positive periodic solution which is globally asymptotically stable.In Chapter 3, we investigate a three-species mixed system with HollingⅢfunctional response and stage-structure for prey. The main conclusion are given byTheorem 3.1 System (3.1) is permanent, if all coefficients satisfyTheorem 3.2 Assume that condition (H1) holds. Then there is a positiveωperiodic solution of system (3.1).Theorem 3.3 Assume that condition (H1) and are satisfied. Then system (3.1) has a unique positiveω-periodic solution which is globally asymptotically stable.In Chapter 4, we consider a noautonomous competition delay system with HollingⅢfunctional response and diffusion for predator. Initial value condition xi ( s ) =φi( s), s∈[ ?τ,0],φi(0) > 0, i= 1,2,3,4. (H1)Theorem 4.1 Assume system (4.1) satisfies the following a set of conditions where.Then system (4.1) is permanent,Theorem 4.2 Assume that system (4.1) satisfies condition (H1)and (H3) and the following conditionsThen system (4.1) has a unique positive almost periodic solution P (t ) which is globally asymptotically stable. When system (4.1) isω-periodic system, the corresponding result is given.