Some Ecological Models With Diffusion And Cross-diffusion

In this paper some prey-predator models and eco-epidemiology models arestudied. Effect of diffusion and cross-diffusion on the stability of constantsolutions,existence and non-existence of non-constant steady state solutionsis mainly considered. Global existence and blow-up of positive solutionsfor a strongly coupled and degenerate quasilinear parabolic system for amutualistic model are also studied. This thesis consists of the following fiveparts.After briefly reviewing the history of the development of biomathematics,especially the history of dynamics model of biological groups,epidemiologymodels and eco-epidemiology models, in the introduction we give the mainresults of this thesis and the direction we will investigate next.To our knowledge, all the eco-epidemiology models are ODE systems. Wewill study two reaction-diffusion systems for eco-epidemiology models. Inthe first section of Chapter one, we consider the following eco-epidemiologymodel: where d1,d2,d3 are diffusion coeffcients, r,a,k and mi(i = 1···5) are posi- tive constants. We can rewrite (1) as Ut = D U +F(U). The ODE systemof (1) was proposed by J. Chattopadhyay and N. Bairagi in 2001 whenthey studied the relationship of the susceptible Tilapia population, infectedTilapia population and their predator Pelican birds population in Salton seaof California. They gave the dissipation, persistence of the ODE system andthe stability of constant solutions. We consider (1) in a bounded domain∈RN with smooth boundary with homogeneous Neumann boundary con-dition and non-negative initial value. Using analytic semigroup theory, wegive the dissipation of (1). Subsequently, we give some a priori estimatesof upper bound and the Harnack Inequality of the positive steady state so-lution of (1). Using these estimates and Poincar′e Inequality, we have thefollowing theorem:Theorem 1 Suppose constant d satisfy 0 < d min{d1,d2,d3}, thenthere exist positive constants Cd2 = Cd2(Λ, ,d,d1),Cd3 = Cd3(Λ, ,d,d1,d2)such that (1) has no non-constant positive steady state solution, when d2Cd2 or d3 Cd3, whereΛ= (r,a,k,m1,···,m5).Since we can't give the uniformly lower bound, we can not prove theexistence of non-constant steady state solutions by degree theory directly.Here we use the bifurcation method proposed by M. X. Wang to prove thenon-constant steady state solutions can bifurcate from constant solution.Using Sp to denote the positive spectrum of on with the homogeneousNeumann boundary condition, that is, Sp = {μ1,μ2,μ3,···}. Define non-constant steady state solutions of (1).In this section, we find that non-constant positive steady state solutionsexist when d2 is in a suitable range.In the second section of Chapter one, we study a eco-epidemiology modelin which the functional response of predator is v/(aw+v), namely, we changethe functional response v/(a+v) of the first section to v/(aw +v). Y. Xiaoand L. Chen studied the systems without di usion. They found that thesystem is rich in boundary dynamics. Here we consider the inner dynamicsof the system (stability of positive constant solution) and the e ect of di u-sion on the existence of non-constant positive steady state solutions. Similarto the first section, we give the dissipation of this system and stability ofthe constant solution of the ODE system. Under some conditions, we givea priori estimates of upper bound and lower bound and Harnack Inequality.Using this estimates, we find that the system has no non-constant steadystate solution when d2 is large enough. Subsequently, using degree theory,we prove that non-constant positive steady state solutions exist when d2,d3satisfy some conditions and d1 is large enough. Finally, using d3 as thebifurcation parameter, we consider the bifurcation of non-constant positivesteady state solutions of the system.We find that the e ect of di usion on the existence of non-constant pos-itive steady state solution is di erent for di erent functional response.In Chapter two, we discuss the following strongly coupled predator-preysystem: with homogeneous Neumann boundary condition and non-negative initialvalues. The cross-di usion term is that consumers move upward along gra-dients of prey density.First of all, we consider the case d3 = 0. Using the the comparisonprinciple of parabolic equation, we give a condition of the dissipation andpersistence property of (2). Using the method proposed by M. X. Wang,we prove that the constant solution of (2) is locally asymptotically stablewhenBy constructing a Lyapunov functional, we prove the following global asymp-totically stable theorem:Theorem 3 hold, then the constantsolution U of (2) is global asymptotically stable. Especially,(2) has nonon-constant positive steady state solution.Next, we consider the e ect of d3. Some a priori estimates of upper andlower bound independent of d3 are given. Using the estimates and Poincar′eInequality and Young's inequality withε, we can prove the following theo-rem about non-existence of non-constant positive steady state solution:Theorem 4 Let (a,b,c,α,δ,β)∈(0,∞)6 be given. For any positive con-stantε, there exist positive constants D1,D2 depending on (a,b,c,α,δ,β),εand such that (2) has no non-constant positive steady state solution whend1 D2(d2) and d2 D1.Finally, using degree theory, we prove that non-constant positive steadystate solutions exist when d1,d3 satisfy some conditions and d2 is largeenough or for arbitrary d2 when d1 satisfy some conditions and d3 is largeenough. In this chapter, we find that the constant solution is locally asymptoticallystable without cross-diffusion, non-constant steady state solutions can befound when cross-diffusion occurs.In Chapter three, we study the following strongly coupled prey-predatorsystem with homogeneous Neumann boundary condition and Holling IIfunctional response:We have the following theorem about stability:Theorem 5 Let r,s be positive constants, positive constants q,αsatisfyα1, 1+α< q orα> 1, 1+α< qα(α+1)/(αff1), then the positiveconstant solution is asymptotically stable with respect to the dynamics (3).Theorem 6 If d3 = d4 = 0,q >α+ 1,qα(qα), then the positiveconstant solution of (3) is globally asymptotically stable, this implies that(3) has no non-constant positive steady state solution.Subsequently, we give some a priori estimates to the positive solutions of(3) which depend on min{1 + d3,1 + d4}. Denotewhere A11 = . Using a priori estimates and degree theory, wehave the following theorem about the existence of non-constant steady statesolution:Theorem 7 Assume thatα,q satisfy and (5). (a) Suppose d1,d3,d4 are given such thatΛ2∈(μj,μj+1) for some positiveeven integer j, then there exists a positive constant d2 such that (3) has atleast one non-constant positive steady state solution when d2 > d2 .(b) Suppose d1 and d3 are given such thatΛ4∈(μj,μj+1) for some positiveeven integer j. Then for any given d2 > 0, then there exists a positiveconstat d 4 such that (3) has at least one non-constant positive steady statesolution when d4 > d4 .However, we can not use the degree theory to prove the existence of non-constant steady state of (3) when d3 is large enough. At the end of thischapter, we give a condition of existence of non-constant steady state of (3)when d3 is large enough by the singular perturbation theory.Theorem 8 Let (α,k,q,d2,d4)∈(0,∞)4×[0,∞) be given. Ifhas only zero solution, where F(w) = w + d4w2/(k + w), G(w) =(w/d2)[q/(w+α) 1], then, for every d1 > 0, there exists a positive constantε0 such that (3) has at least one non-constant positive steady state solutionwhen d3ε10.From Theorem 6 and Theorem 7 we can conclude that the constant solu-tion is globally asymptotically stable without cross-di usion, non-constant steady state solutions can be found when cross-di usion occurs. This resultis stronger than the result we conclude in Chapter two.In Chapter four, global existence and blow-up of solutions for a degenerateOwing to the degeneracy of (6), we use the method of regularization toconstruct a sequence of approximation solutions to obtain the local existenceof positive classical solution of (6). Let T be the maximal existence timeof the ODE system:We have the following local existence theorem:Theorem 9 System (6) has a positive classical solution (u,v) with u,v∈C2,1( )×(0,T )∩C(×[0,T )).Comparing the solution of (6) with the solution of an ODE system, wehave the global existence theorem:Theorem 10 If b1c2 > b2c1, then System (6) has a global solution (u,v)which is uniformly bounded in×[0,∞). Strongly coupled, homogeneous Dirichlet boundary condition and havingabsorption terms b1u2 and c2v2, it is more complex to prove that the solutionof (6) blows up. Generally, the comparison principle no longer holds for thestrongly coupled system, so we can not find a lower solution which blowsup. Here we setwhereλ1 is the first eigenvalue of ?? on ? with homogeneous Dirichletboundary condition, constantθ> 0 satisfies(4 +λ1)θ+ 3θ2 +λ1 < min{(c1 ? b1)/α,a1/(1 ?α),(b2 ? c2)/β,a2/(1 ?β)}.Using reduction to absurdity, we have the following theorem:Theorem 11 If b1 < c1,c2 < b2 andα,β1, then the solution of (6)blows up provided thatThis theorem implies that the solution of System (6) blows up when thetwo species are strongly mutualistic and the habitat is large enough.