Existence And Properties Of Positive Solutions To Multi-Species Prey-predator Models

This paper is a survey on the studies of multi-species prey-predator models. In this paper, our main interest is to study a class of prey-predator problem. We focus on the existence and non-existence, the local and global asymptotic stability of positive solutions to prey-predator models as well as some specific applications. The paper is divided into four chapters. The main contents are as follows:In chapter one, we introduce a survey to the development of the study to multi-species prey-predator models in biological mathematics. We also give the outline to this paper and state the main results we will introdue.In chapter two, we introduce some models interacting between the predator species and the prey species. First, we list some results to the models of ecosystems with constant coefficiets and variable coefficents. The results include the existence and non-existence results and the asymptotic stability of positive solutions , which were obtained by using Lyapunov-Schmidt procedure. The problems are AndThe main results obtained are as follows:Theorem 1. Forα>λ₁ =λ₁(0),there exists a positive solution for (1) if the following condition is satisfied:where it is understood thatθ_b≡0 for b≤λ₁,Forα≤λ₁ =λ₁(0), (1) has no positive solution.Theorem 2. For almost every (β, b, d)∈O₂, there exists a small positive constantδ> 0 such that d₁/d₂≤δ, then all steady-state solutions of (1) onГ_2j-1²(j=1,2,…[(k +1)/2]) are asymptotically stable in the topology of X , while all steady-state solutions of (1) onГ_2j²(j= 1,2,…[k/2]) are unstable.Theorem 3. Let (k,ρ(x), c(x), d(x)) be an arbitrarily fixed vector. And there exists a positive continuous function b = (?)(a)(a≥0) with (?)(0) = 0 and (?)(a) =∞such thatThen (2) has no positive solution.Next, we introduce some results including the existence and non-existence of positive solutions to the models by applying the degree theory, the bifurcation theory, the linearized stability theory and the spectral analysis theory,2-species prey-predator models are studied by many authors. we will provide the basis for the main results of the following models:First, we consider (3) with d₃ = d₄ = 0 and our main results are :Theorem 4. There exists at least one positive solution to (3), if one of the following conditions is satisfied:Theorem 5. Let b > d₂λ₁,there exists a positive constantδ₀ > 0 such thatμ₁(mθ_b/d₁)d₁< a <μ₁(mθ_b/d₁)d₁+δ₀,then (3) has a unique positive solution and this solution is locally linearly stable.After that, we list some results for non-constant positive solutons, when d₃≠0, d₄≠0, andα= 0, Further, we may prove that Problem (3) has a unique positive constant solution, written as (?) = ((?), (?))^T. For the results of the existence of no non-constant positive solutions, we haveTheorem 6. Let D₁,D₂,D₃,D₄ be arbitrarily positive constants while D₁and D₂ are relatively small and D₃ and D₄ > (?) are relatively big.(1) there exist positive constants (?)₂ and (?)₄ depending on D₁, D₂,D₃,D₄, satisfing(?)₂ > D₂,(?)₄ < D₄, such that d₁≥D₁, d₂ > (?)2,d₃≤D₃,d₄≤(?)₄,then (3) has no non-constant positive solution. (2) there exist positive constants (?)₁ and (?)₃ depending on D₁, D₂, D₃, D₄, satisfing (?)₁> D₁,(?)₃ < D₃,such that d₁ > (?)₁,d₂≥D₂,d₃ < (?)₃, d₄≤D₄,then (3) has no non-constant positive solution.In general, nothing will never change and hence the assumption that the coefficients are constants is not realistic. Therefore, we next introduce some results of the existence and the stability of positive solutions to some models with variable coefficents. Also, we introduce some results which reflect the effects of functional response and transform ratio to the solutions.In chapter three, we introduce some results for positive solutions to interacting systems of multi-species in all possible patterns of interactions in terms of the combinations of competition, symbiosis, and predation under the homogeneous Robin-Dirichlet boundary conditions.In section one, we state a basic fact of the degree theory. Secondly we list some results such as the existence and non-existence of 3-species prey-predator models, and simultaneously point out the difference between 2-species and 3-species prey-predator models of ecosystems which are different in inhomogeneous function's expression and the conditions the models satisfied. Finally, we state some results obtained by using upper-lower solutions technique.The model we will consider is:where d_i,a_i and b_ij(i,j = 1,2,3) are positive constants.For simplicity of notations, we denote, respectively k₁(u,v, w) = a₁ - b₁₁u - b₁₂v - b₁₃w, k₂(u,v, w) = -a₂+b₂₁u-b₂₂v-b₂₃w, k₃(u,v, w) = -a₃-b₃₁u+b₃₂v-b₃₃w, K(u,v, w) = (k₁,k₂, k₃). For above systems, the main results are:Theorem 7. If b₁₁b₃₃≥(b₁₂b₃₂/4b₂₁b₂₃)((b₂₁/b₁₂)b₁₃ + (b₂₃/b₃₂)·b₃₁)², and the equation K(U) = 0 has a unique positive root U^* = (u^*,v^*,w^*). then U^* is the only solution of (4).Theorem 8. There exist positive constants C_d1 = C_d1(a_i,b_ij,λ,d₂,d₃) and C_d3 = C_d3(a_i,b_ij,λ,d₁,d₂) such that the system (4) has no non-constant solution provided that d₁≥C_d1 or d₃≥C_d3. Moreover, suppose that d₁> (α₁/λ),d₃ > (b₃₂/λ)(a₁b₂₁/b₁₁b₂₂) - (a₃/λ) and 4(d₁λ- a₁)(d₃λ+ a₃ - b₃₂(a₁b₂₁/b₁₁b₂₂)) > a₁²((b₁₃/b₁₁) + (b₃₁b₃₂b₂₁/b₁₁b₂₂b₃₃))² hold. Then there exists a positive constant C_d2 = C_d2(a_i,b_ij,λ,d₁,d₃) such that the system (4) has no non-constant solution provided that d₂≥C_d2.Furthermore, we also introduce the global existence and blowup results of the solutions of a pababolic system describing 3-species in cooperating model.In section two. we introduce some results of multi-species prey-predator models. The more species there are, the more complex their relations get, which not only influence directly the existence, non-existence and stability of positive solutions of ecosystems arising in prey-predator models, but also bring some difficulties to our study. Hence, only a few people study such problems. We may further investigate such problems in the future.In chapter four, we state some specific applications in agriculture, chemistry, medicine of prey-predator moedls. For example, Frank explained "Hyperpredation process" by analysing prey-predator moedls, the DDT restriction to insects for cotton aphid and ladybug in USA, the edible fish and shark in Mediterranean, leopard cats and American rabbits in Canddian forests, etc, which are typical of such survival method. Subsequently, we introduce some hot embranchments of models of ecosystems: Nonlinear Dynamic, Temporal and Spatial Dynamic of Species. In the final part of the article, we propose that the existence, non-existence and the uniqueness of periodic solutions to multi-species prey-predator models and the existence and stability of positive solutions to multi-species prey-predator models with variable coefficent should be considered to be a very interesting topics and with great prospects.