| Population ecology is a branch of science to study the development law of the species. By establishing the species dynamics models, we can describe the interaction among species or between species and the environment. Through understanding, explaining and predicting the mathematical models, we can also analysis the change of each species, so as to manage and protect the bio-logical species effectively. Epidemiology can reflect the dynamic characteristics of the infection according to the species growth characteristics, the occurrence of diseases, the spread of diseases and the changes within population. It is to reveal the spread law of the infection by analyzing the dynamic properties of the mathematical models, so as to prevent and control the infectious dis-eases effectively. Therefore many scholars study the dynamics behavior of the population ecology and epidemiology by establishing various forms of popula-tion models and epidemic models. In this dissertation, we mainly study the dynamics properties of some biology models and epidemic models.This dissertation is divided by four chapters. The main contents and the theorems are as follows.In the first chapter, we mainly introduce the background and significance of predator-prey models and epidemic models, the basic theories of fractional differentiation, stochastic process and Markov switching process.In the second chapter, we discuss a fractional order predator-prey system with Holling Type II function response and a fractional order SIR model, respectively.Firstly, we consider a fractional order predator-prey system with Holling Type II function responsewhere x(t) and y(t) represent the population densities of prey and predator at time t, respectively. The parameters a, b/a,γ,β, e and k are positive constants that stand for prey intrinsic growth rate, carrying capacity, the maximum ingestion rate, half-saturation constant, predator death rate and the conversion factor, respectively. Some complexity on multiscale analysis can be simplified by establishing the fractional order differential equations. A unique positive solution of the system (0.0.12) is obtained by using the theory of the modified Riemann-Liouville derivative.Theorem 0.0.1 For any initial value (x(0), y(0)) ∈ R+2, there is a unique global solution (x(t),y(t)) of system (0.0.12) on t≥0.In order to prove the asymptotical stability of the positive equilibrium of system (0.0.12), we study the Lyapunov stability theory of the fractional order differential equation, firstly. Consider the fractional order differential equationHere x and f(x) are n-dimensional column vectors. Suppose that f(0)= 0, f(x) is continuous in G:|x|< H and satisfies local Lipschitz condition. We get the following theorem.Theorem 0.0.2 Let V (x) be a C1- function satisfying V(0)= 0, V(x)>(1) If dαV/dtα≤ 0, then system (0.0.13) has a stable null solution;(2) If dαV/dtα< 0, then system (0.0.13) has an asymptotically stable null solution;(3) the null solution of system (0.0.13) is unstable provided dαV/dtα> 0.Using this theorem, we discuss the stability of the positive equilibrium of model (0.0.12) and get the following theorem.Theorem 0.0.3 If akγ> aeγ+be such that system (0.0.12) has a positive equilibrium (x*,y*). Then system (0.0.12) is asymptotically stable.Secondly, we consider a fractional order SIR modelwhere S(t),I(t),R(t) represent the number of the individuals susceptible to the disease, of infected members and of members who have been removed from the possibility of infection through full immunity, respectively. The parameter A,γ, μ, ε,γ are positive constants that stand for the influx of individuals into the susceptibles, the disease transmission coefficient, the natural death rates, the additional dead-rate constant due to the disease suffered by the individuals in I(t), the rate of recovery from infection, respectively. The basic reproduc-tion number Rq=β∧/μ(μ+ε+γ) is the threshold of the system for an epidemic to occur. In order to study the existence and uniqueness of the solution of mod-el (0.0.14), we give the general solution of the fractional differential equation system, firstly. Consider the fractional differential equation where A(t) and f(t) are functions defined in I ∈ R1.Theorem 0.0.4 If f(t)=≠0, then is the general solution of the fractional differential equation (0.0.15).Using the theory of modified Riemann-Liouville derivative, Lyapunov method and the Theorem 0.0.2, we prove the existence and uniqueness of the solution and the stability of the equilibrium of model (0.0.14). The main conclusions are as follows.Theorem 0.0.5 For any given initial value is a unique global positive solution (S(t),I(t), R(t)) of model (0.0.14) ont> 0.Theorem 0.0.6 such that system (0.0.14) has the disease-free equilibrium Then system (0.0.14) is asymptotically stable.Theorem 0.0.71 If such that system (0.0.14) has the endemic equilibrium E*, then system (0.0.14) is asymptotically stable.In the third chapter, we study a stochastic SEIR epidemic model with saturated incidence rate driven by Levy noiseConsider some diseases may infect in latent period, we divide the population into the susceptible individuals (S), the latent individuals (E), the infective in-dividuals (I) and the recovered individuals (R). The parameters μS,μE, μI,μR represents the natural death rates of S, E, I and R, respectively. λ is the birth rate.β is the per eapita contact rate.The average duration 0f the latent state is 1/θ.δ is the additional disease caused rate suffered by the infectious individu-als.γ is the recovery rate of infeetious individuals.Di(t)>-1(i=1,…,4), Brownian motion Bi(t)(i=1,…,4)is given on a complete probability space (Ω,F,Ρ)with a filtration {F)t>0.σi>0(i=1,…,4) is the intensity of Bi(t)(i=1,…,4).N(dt,dy) is Poisson measure and v(dy) dt is the stationary compensator.v is defined on a measure subset A={y|y|<r,r∈[0,∞)} satisfying v(A)<∞.Suppose that for each c>0,there exists Lc>O such that (H1) ∫A|Fi(x,y)-Fi(z,y)|2v(dy) ≤ Le|x-z|2,(i= 1,... ,4),where F1(x,y) = D1(y)S(t), F2(x,y) = D3(y)E(t), F3(x,y) = D3(y)I(t),F4(x,y) = D4(y)R(t), |x| ∨|z| ≤ n. (H2) |log(1 + Di(y))|v(dy) < ∞, (i= 1,... ,4).In these assumptions, we get the existence and uniqueness of the solution of model (0.0.17).Theorem 0.0.8 Let the assumption (H1) and (H2) hold, for any given initial value (S(O),E(O),I(O),R(O)) ∈R+4, the system (0.0.17) has a unique global solution (S(t),E(t),I(t),R(t)) ∈ R+4 for all t ≥ 0 almost surely.Using the Lyapunov method, we investigate the asymptotic behavior of the solution of stochastic model (0.0.17) around the disease-free equilibrium point and the endemic equilibrium point of its deterministic model. We get the following main conclusions.Theorem 0.0.9 Let the assumption (H1) and (H2) hold. If R0 < 1 and satisfied the following conditions:then for any given initial value (S(0),E(0),I(0),R(0)) ∈ R+4, the solution (S(t),E(t),I(t),R(t)) of system (0.0.17) has the following property:Theorem 0.0.10 Let the assumption (HI) and (H2) hold. If R0> 1 and satisfied the following conditions:then for any given initial value (S(0),E(0),I(0),R(0)) ∈ R+4, the solution (S(t),E(t),I(t),R(t)) of system (0.0.17) has the following property:In the fourth chapter, we study a hybrid stochastic SIR model driven by Levy noisewhere S(t),I(t),R(t) represent susceptible individuals, infective individuals and recovered individuals, respectively. The parameter ∧,γ,μ,ε ,γ are posi-tive constants that express the influx of individuals into the susceptibles, the disease transmission coefficient, the natural death rates, the additional dead-rate constant due to the disease suffered by the individuals, the rate of recovery from infection, respectively. Bi(t) is standard Brownian motion defined on a complete probability space (Ω,ζ, P) with a filtration{Ft}t≥0 satisfying the usual conditions.σi> 0 represents the intensity of Bi(t). Di(y)>1, N is the compensated Poisson random measure given bywhere v(dy)dt is the stationary compensator and u is defined on a measure subset M={y||y|< r, r ∈ [0,∞)} satisfying v (dy)< 00. In order to prove the existence and uniqueness of the solution of model (0.0.20), we assume the jump diffusion coefficients satisfy that for each n> 0, there exists Ln such thatWe get the following Theorem:Theorem 0.0.11 Let the assumption (A1) and (A2) hold, for any given initial value (S(0),I(0),R(0)) ∈ R+3, the system (0.0.20) has a unique global solution (S(t),I(t),R(t)) ∈ R+3 for all t≥0 almost surely.When the model (0.0.20) is not affected by Levy noise, the corresponding hybrid SIR model has the threshold Rr0> 1, there exists the endemic equilibrium Er*(Sr*,Ir*,Rr*). Using the Lya-punov function Ito formula and the Gronwall inequality, we investigate the asymptotic behavior of model (0.0.20) around the disease-free equilibrium Er0 and the endemic equilibrium Er*. The main conclusions are as follows.Theorem 0.0.12 If Rr0<1, and satisfied thatthen for any given initial value (S(0),1(0), R(0)) ∈ R+3, the solution (S(t),I(t), R(t)) of system (0.0.20) has the following property:Theorem 0.0.13 If Rr0> 1, and satisfied thatthen for any given initial value (5(0),1(0), R(0)) ∈R+3, the solution (S(t), I(t), R(t)) of system (0.0.20) has the following property:... |
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