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... The dynamics and oscillation in logistic differential equations with or without time delays have been the object of intensive analysis by numerous authors [2][3][4][5][6]8,9,12,13] . The follow ing reaction-diffusion system describes the evolution of a population u subject to diffusion, having "self-limitation" in growth, with spacially varying environmental parameters: We assume that n is a bounded domain in Rn with smooth boundary an. ...
... The functions a(x) and h(x) are Holder continuous on IT, with a(x), h(x) > 0 in n. The differential operator A is defined as D(A) = { f E Co (IT) ; fE n w 2 , p (n); Af E Co (IT) } , (2) p >l (3) where all the coefficients Ctij and (Jj are Holder continuous in x. A is uniformly strongly elliptic, that is, there is a constant d > ° such that n n L Ctii (x)�i�i � d L �;' x E n, (6," "� n) ER n . ...
... where n is any nonnegative integer. In this way we have r E C ( [O, 00], [1,3]). ...
Key words: Diffusive logistic equations, time-varying delay effects, stability and asymptotic behavior. Abstract: In this paper we study the global stability in a reaction-diffusion model for single species population growth with spatially varying environmental coefficients and time-varying delays. It is shown that as long as the magnitude of the instantaneous self-limitation effect is larger than that of the delay effects, the solution of the reaction-diffusion system has the same asymptotic behavior (extinction or converging to the positive steady-state solution) as in the case without time delay. Numerical simulations for both cases (with or without time delay) are demonstrated fo r the purpose of comparison.
... Periodic semi-linear reaction diffusion equations are of particular interests since they can take into account seasonal fluctuations occurring in the phenomena appearing in the models, and have been extensively studied by many researchers (of [1,2,4,7,12,17,[24][25][26]28,29]). At the same time, much attentions have also been paid to periodic semi-linear parabolic equations with delays where time lag is taken into consideration in nonlinear reaction functions (of [11,21,27,31,[33][34][35]). In order to model a more general way of spreading behavior in space, the use of degenerate parabolic operators has been proposed. ...
... In final section, as the application of main results in previous section the sufficient conditions are given for the existence of nontrivial nonnegative periodic solutions of the delayed logistic equation with the nonlinear diffusion, and so called coexistence periodic solutions of the delayed degenerate parabolic system which models the competitor-competitor-mutualist system, respectively. In the case of linear diffusion, these models were studied by many researchers (of [10][11][12][15][16][17]27,28,37]). Throughout this paper we make the following assumptions: ...
... In the case of m = 1, Eq. (4.1) reduces to be semi-linear parabolic equation and has been investigated by many authors (of [11,27,37]). In this situation, the sufficient conditions to ensure the existence of nontrivial periodic solutions are usually given in terms of the periodic parabolic principal eigenvalue. ...
This paper is concerned with a class of periodic degenerate parabolic system with time delays in a bounded domain under mixed boundary condition. Under locally Lipschitz condition on reaction functions, we apply Schauder fixed point theorem to obtain the existence of periodic solutions of the periodic problem. With quasi-monotonicity in addition, we also show that the periodic problem has a maximal and a minimal periodic solutions. Applications of the obtained results are also given to some nonlinear diffusion models arising from ecology.
... The study of the logistical equation over the decades has been the subject of a study for many differential purpose [12,14,19,20] and references therein. In the context of differential equation (integer order), it has also been considered, see for example [8,15,18] and references therein. Gopalsamy [17] had his research directed at the asymptotic behavior of non-constant solutions of delay logistic equations. ...
... Gopalsamy [17] had his research directed at the asymptotic behavior of non-constant solutions of delay logistic equations. Feng and Lu [18], also dedicated to investigating the asymptotic periodicity in diffusive logistic equations with discrete delay, i.e., as follows u(x, 0) = u 0 (x) for (t, x) ∈ (0, ∞) × Ω, was proved by Schiaffino [22] and Yamada [23]. ...
... where p > 1, τ ¿0, T 0 β(s)ds = 0, e(.) is T −periodic and T 0 e(s)ds = 0. At the same time, much attentions have also been paid to periodic semi-linear parabolic equations with delays where time lag is taken into consideration in nonlinear reaction functions, see for example, the work of Feng and Lu in 1996 [45], Lu and Feng in 1996 [96]; Feng and Lu in 1999 [46], Pao in 2005 [107], Wang et al. in 2006 [140]; Wang in 2008 [139], Yang et al. in 2010 [148] and Wang and Yin in 2011 [143]). ...
In this survey, we collect a few results scatted in the literature covering advances in systems of periodic delay differential equations including both modeling and theory. We will present the (almost) most oldest and (almost) most recent contributions made for this subject.
... See e.g. [7,9,11,20,[26][27][28]. We refer [7] to a systematic analysis on memory effects in populational dynamics. ...
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... (1) In a particular case of our food-limited model with b(t, x) ≡ d(t, x) ≡ 0, the result Theorem 4.1 is coincident with that in [3] for the case τ = mT (m ∈ N ) corresponding to the Logistic model. ...
... The boundary condition (1.6) means that the habitat Ω is surrounded by a lethal environment. There is a large literature relating to periodic semi-linear parabolic equations with delay; see for example [8,14,24]. On the other hand, over the past decades degenerate parabolic equations have been the subject of extensive study, and much progress has been made on the existence, uniqueness, regularity properties of generalized solutions, and other phenomena such as finite speed of propagation of perturbation, localization and blow up; see [5,31,34] for an overview and extensive bibliographies. Recently, considerable attention has been attracted to the periodic problem for degenerate parabolic equations without delay because of connections with fluid flow in porous media and population dynamics; see for example [1,3,21,33,42] and references therein. ...
This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.
... the past twenty years, reaction-diffusion systems derived from multi-species interactions (competition, predator-prey, and food chain) have been extensively studied ([1, 2, 3, 4, 6, 8]). Recently, many authors also worked on population models with time delays (see, for example, [5, 7, 11, 14, 15, 16]) and obtained some results on global existence of a solution, stability or instability of the steady states, periodicity, and positive global attractors. In this paper we extend the study of population dynamics from models with two or three species to a model of four-species food chain with diffusion and delays. ...
We study a reaction-diffusion system modeling the population dynamics of a four-species food chain with time delays. Under Dirichlet and Neumann boundary conditions, we discuss the existence of a positive global attractor which demonstrates the presence of a positive steady state and the permanence effect in the ecological system. Sufficient conditions on the interaction rates are given to ensure the persistence of all species in the food chain. For the case of Neumann boundary condition, we further obtain the uniqueness of a positive steady state, and in such case the density functions converge uniformly to a constant solution. Numerical simulations of the food-chain models are also given to demonstrate and compare the asymptotic behavior of the time-dependent density functions.
... In this section, numerical solutions in both systems are obtained, with a > b and a < b, and demonstrated for the purpose of comparison. By discretizing the differential equation systems into finite-difference systems, we obtain numerical solutions through the monotone iterative scheme developed and employed in several earlier papers[1, 2, 9, 10, 12]. The principal eigenvalue of the Laplacian operator on Ω = (0, 1) with zero Dirichlet boundary conditions is λ 0 = π 2 . ...
In this paper we study a class of reaction-diffusion systems modelling the dynamics of "food-limited" populations with periodic environmental data and time delays. The existence of a global attracting positive periodic solution is first established in the model without time delay. It is further shown that as long as the magnitude of the instantaneous self-limitation effects is larger than that of the time-delay effects, the positive periodic solution is also the global attractor in the time-delay system. Numerical simulations for both cases (with or without time delays) demonstrate the same asymptotic behavior (extinction or converging to the positive T-periodic solution, depending on the growth rate of the species).
... (1) In a particular case of our food-limited model with b(t, x) ≡ d(t, x) ≡ 0, the result Theorem 4.1 is coincident with that in [3] for the case τ = mT (m ∈ N ) corresponding to the Logistic model. ...
In this paper, a general reaction–diffusion food-limited population model with time-delay is proposed. Accordingly, the existence and uniqueness of the periodic solutions for the boundary value problem and the asymptotic periodicity of the initial-boundary value problem are considered. Finally, the effect of the time-delay on the asymptotic behavior of the solutions is given.
... The logistic delay differential equation as a model of single-species population growth has been considered in [13,14]. Nonlinear periodic diffusion equations arise naturally in population models [15] where the birth and death rates, rates of diffusion, rates of interactions and environmental carrying capacities are periodic on seasonal scale. The existence and global stability of a T-periodic solution of periodic boundary-value problem of the logistic model has been studied by Hess [16]. ...
In this paper, the competitor–competitor–mutualist three-species Lotka–Volterra model is discussed. Firstly, by Schauder fixed point theory, the coexistence state of the strongly coupled system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak. Secondly, the existence and asymptotic behavior of T-periodic solutions for the periodic reaction–diffusion system under homogeneous Dirichlet boundary conditions are investigated. Sufficient conditions which guarantee the existence of T-periodic solution are also obtained.
... The model problem (4.1) has been studied by many investigators and various qualitative properties of the solution have been obtained (e.g., see121314). Here we discretize (4.1) into a finite-difference system and construct a pair of upper and lower solutions so that the monotone iterative schemes given in Section 3 can be used to computer numerical solutions. ...
The aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction-diffusion equations with time delays. The iterative schemes lead to computational algorithms as well as existence, uniqueness, and upper and lower bounds of the solution. An application to a diffusive logistic equation with time delay is given.
Population dispersal
plays an important role in the population dynamics which arises from environmental and ecological gradients in the habitat. We assume that the systems under consideration are allowed to diffuse spatially besides evolving in time. The spatial diffusion arises from the tendency of species to migrate towards regions of lower population density where the life is better. The most familiar model systems incorporating these features are reaction diffusion equations.
In the Lotka-Volterra cooperating reaction-diffusion system if the diffusion coefficients are constants then for a certain set of reaction rates in the reaction function the solution of the system blows up in finite time, and for another set of reaction rates, a unique global solution exists and converges to the trivial solution. However, if the diffusion coefficients are density-dependent then the dynamic behavior of the solution can be quite different. The aim of this paper is to investigate the global existence and the asymptotic behavior of the solution for a class of density-dependent cooperating reaction-diffusion systems where the diffusion coefficients are degenerate. It is shown that the time-dependent problem has a unique bounded global solution, and in addition to the trivial and semi-trivial solutions the corresponding steady-state problem has a positive maximal solution and a positive minimal solution. Moreover, the time-dependent solution converges to the maximal solution for one class of initial functions, and to the minimal solution for another class of initial functions. The above convergence property holds true for any reaction rates in the reaction function. Applications of the above results are given to a porous medium type of reaction-diffusion problem as well as other types of diffusion coefficients, including the finite sum and products of these functions.
In this paper, the stability of periodic solution is investigated for a class of periodic Logistic equation with delay using the Floquet theory of linearized stability of periodic solution.
Environmental variation plays an important role in many biological and ecological dynamical systems. This monograph focuses on the study of oscillation and the stability of delay models occurring in biology. The book presents recent research results on the qualitative behavior of mathematical models under different physical and environmental conditions, covering dynamics including the distribution and consumption of food. Researchers in the fields of mathematical modeling, mathematical biology, and population dynamics will be particularly interested in this material. © 2014 Springer International Publishing Switzerland. All rights are reserved.
A linearized compact difference scheme is presented for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. The unique solvability, unconditional convergence and stability of the scheme are proved. The convergence order is O(τ2+h4)O(τ2+h4) in L∞L∞ norm. Finally, a numerical example is given to support the theoretical results.
This paper deals with a two species model with Schoener’s competitive interaction. The existence and the asymptotic behavior of T-periodic solutions for the periodic system of quasilinear parabolic equations under nonlinear boundary conditions are given by using upper and lower solutions and corresponding iteration. The numerical simulations are also presented to illustrate our result. It is shown that periodic solutions may exist if the inter-specific competition rates are weak.
This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem as well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm.
As experimental results reveal, the growth rates, diffusion rates and interaction coefficients of biological species are very sensitive dependent variables of external environmental conditions. A slight change in those conditions may effect the whole evolution process. Nonlinear periodic diffusion equations arise naturally in population models [l-5] where the birth and death rates, rates of diffusion, and environmental carrying capacities are periodic on a seasonal scale. The following periodic-parabolic logistic boundary value problem describes the evolution of a population u subject to diffusion, having “self-limitation” in growth, with periodically varying environmental parameters ~
In this paper, the cooperating two-species Lotka–Volterra model is discussed. The existence and asymptotic behavior of T-periodic solutions for the periodic reaction diffusion system under homogeneous Dirichlet boundary conditions are first investigated. The blowup properties of solutions for the same system are then given. It is shown that periodic solutions exist if the intra-specific competitions are strong whereas blowup solutions exist under certain conditions if the intra-specific competitions are weak. Numerical simulations and a brief discussion are also presented in the last section.
Existence of maximal and minimal periodic solutions of a coupled system of parabolic equations with time delays and with nonlinear boundary conditions is discussed. The proof of the existence theorem is based on the method of upper and lower solutions and its associated monotone iterations. This method is constructive and can be used to develop a computational algorithm for numerical solutions of the periodic-parabolic system. An application is given to a competitor-competitor-mutualist model which consists of a coupled system of three reaction-diffusion equations with time delays.
The existence of periodic quasisolutions and the dynamics for a couple system of semilinear parabolic equations with discrete time delays and periodic coefficients are investigated using the method of upper and lower solutions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the periodic boundary value system has a pair of coupled T-upper and lower solutions then the periodic boundary value problem has a pair of periodic quasisolutions and the sector between the quasisolutions is an attractor of the delayed periodic parabolic system. Under some additional conditions the periodic quasisolutions are exactly true periodic solutions of the periodic boundary value system. Finally, three models arising from ecology are given to illustrate the obtained results.
The goal of this paper is to study existence of weak periodic solutions for some quasilinear parabolic systems with data measures and critical growth nonlinearity with respect to the gradient. The classical techniques based on C α -estimates for the solution or its gradient cannot be applied because of the lack of regularity and a new approach must be considered. Various necessary conditions are obtained on the data for existence. The existence of at least one weak periodic solution is proved under the assumption that a weak periodic super solution is known.
Continuous time deterministic epidemic models are traditionally formulated as systems of ordinary differential equations for
the numbers of individuals in various disease states, with the sojourn time in a state being exponentially distributed. Time
delays are introduced to model constant sojourn times in a state, for example, the infective or immune state. Models then
become delay-differential and/or integral equations. For a review of some epidemic models with delay see van den Driessche
[228]. More generally, an arbitrarily distributed sojourn time in a state, for example, the infective or immune state, is
used by some authors (see [69] and the references therein).
This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka–Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.
This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.
This paper is concerned with the existence and stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions. The approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for parabolic initial-boundary value problems. A sufficient condition for the stability of a periodic solution is also given. These results are applied to three model problems arising from chemical kinetics, ecology, and population biology.
In this article we study the global stability in reaction-diffusion models for single-species population growth under environmental toxicants with or without time delays. The existence and uniqueness of a positive steady-state solution are established in those models. It is shown that as long as the magnitude of the instantaneous self-limitation and toxicant effects is larger than that of the time-delay effects in the model with delays, the solution of both reaction-diffusion systems has the same asymptotic behavior (extinction or converging to the positive steady-state solution, depending on the growth rate of the species). Numerical simulations for both cases (with or without time delays) are demonstrated for the purpose of comparison.
The aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction–diffusion–convection equations under linear boundary conditions. The boundary-value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method, and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations. The construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function, and the iterative schemes are presented for each of the three types of quasimonotone functions. An application of the monotone iterations, including some numerical results, is given to a modified logistic diffusive equation where the kernel in the integral term may be positive, negative, or changing sign in its domain.
This paper is devoted to a numerical analysis of periodic solutions of a finite difference system which is a discrete version of a class of nonlinear reaction-diffusion-convection equations under nonlinear boundary conditions. Three monotone iterative schemes for the finite difference system are presented, and it is shown by the method of upper and lower solutions that the sequence of iterations from each of these iterative schemes converges monotonically to either a maximal periodic solution or a minimal periodic solution depending on whether the initial iteration is an upper solution or a lower solution. A comparison theorem for the various monotone sequences is given. It is also shown that the maximal and minimal periodic solutions of the finite difference system converge to the corresponding maximal and minimal periodic solutions of the reaction-diffusion-convection equation as the mesh size decreases to zero. Some error estimates between the theoretical and the computed iterations for each of the three iterative schemes are obtained, and a discussion on the numerical stability of these schemes is given. Also given are some numerical results of a logistic reaction diffusion problem.
The existence of time-periodic solutions is proven for a large class of reaction- diusion systems in which Dirichlet boundary data, diusivities, and reaction rates are periodic with common period.
This paper treats a reaction–diffusion system which models the dynamics of three—species food chain interactions in ecology. A sufficient condition is given to ensure the existence of a positive steady—state solution in terms of the natural growth rates of the three species. Under the same circumstance, the reaction—diffusion sysem has a positive global attractor which indicates the permanence effect in the ecological model.
The method of upper-lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions. The idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem. The aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable. These two schemes are modified Jacobi method and Gauss-Seidel method for nonlinear algebraic equations. An advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration. On the other hand, the standard Picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations.
A class of time-delay reaction-diffusion systems which arise from the model of two competing ecological species is discussed. We study the global attractivity of the positive steady states in terms of the natural growth rates of the species. Through a monotone iterative scheme, it is proven that when the natural growth rates are in certain unbounded parameter sets, the time-delay reaction-diffusion system has a positive global attractor enclosed by positive steady-state solutions. Asymptotic stability of the trivial and semi-trivial steady states are obtained explicitly. These persistence and extinction results are also demonstrated numerically.
A linear system of differential equations with the argument [] is studied, where [·] denotes the greatest-integer function. Of special interest are oscillatory and periodic properties of the solutions, which depend on the eigenvalues of a certain matrix associated with the system. Therefore, the results can be extended to more general dynamical systems.
The inequality (∂tu - Δ u)(t, x) ≤ u(t, x)(1 - u(t - τ, x)) is investigated. It is shown that nonnegative solutions of the Dirichlet problem in a bounded interval remain bounded as time goes to infinity, whereas in a more dimensional domain, in general, this holds only if the delay is not too large.
This paper gives necessary and sufficient conditions for global stability of certain logistic delay differential equations for all values of the delay.
Sufficient conditions are obtained for the global attractivity of a positive periodic solution of the delay-logistic equation , where r and K are positive periodic functions of period τ and n is a positive integer; sufficient conditions are also obtained for all solutions to be oscillatory about K.
on considere l'equation differentielle scalaire a retard N'(t)=N(t)(a-bN(t)-N(t-1)). On suppose a,b constantes positives et on etudie les solutions pour t>0 telles que N(t)=N 0 (t), −1≤t≤0, avec N 0 (t) une fonction initiale donnee positive et continue. On obtient des resultats sur la limitation et le comportement asymptotique des solutions
This paper considers a reaction-diffusion system that models the situation in which two predators feed on one prey species. A sufficient condition for the existence of a positive steady-state solution and a positive global attractor is given in terms of the natural growth rates and the relative reaction rates of the three species. We prove that when the natural growth rates are in a certain unbounded parameter set Λ , the corresponding elliptic system has a pair of positive upper-lower solutions which ensures the existence of a positive solution. Two converging monotone sequences are generated from the upper and lower solutions by an iteration scheme and their limits enclose a positive attractor for the reaction-diffusion system. The present paper also gives some explicit information on the limiting behavior of the time-dependent solution in relation to the semitrivial and trivial steady states. Finally, numerical evidence of the long-term coexistence is demonstrated by solving the corresponding finite-difference system.
This paper considers a reaction-diffusion system that models the situation in which two
predators feed on one prey species. A sufficient condition for the existence of a positive
steady-state solution and a positive global attractor is given in terms of the natural
growth rates and the relative reaction rates of the three species. We prove that when the
natural growth rates are in a certain unbounded parameter set , the corresponding
elliptic system has a pair of positive upper-lower solutions which ensures the existence
of a positive solution. Two converging monotone sequences are generated from the upper and
lower solutions by an iteration scheme and their limits enclose a positive attractor for
the reaction-diffusion system. The present paper also gives some explicit information on
the limiting behavior of the time-dependent solution in relation to the semitrivial and
trivial steady states. Finally, numerical evidence of the long-term coexistence is
demonstrated by solving the corresponding finite-difference system.
A basic question in mathematical ecology is that of deciding whether or not a model for the population dynamics of interacting species predicts their long-term coexistence. A sufficient condition for coexistence is the presence of a globally attracting positive equilibrium, but that condition may be too strong since it excludes other possibilities such as stable periodic solutions. Even if there is such an equilibrium, it may be difficult to establish its existence and stability, especially in the case of models with diffusion. In recent years, there has been considerable interest in the idea of uniform persistence or permanence, where coexistence is inferred from the existence of a globally attracting positive set. The advantage of that approach is that often uniform persistence can be shown much more easily than the existence of a globally attracting equilibrium. The disadvantage is that most techniques for establishing uniform persistence do not provide any information on the size or location of the attracting set. That is a serious drawback from the applied viewpoint, because if the positive attracting set contains points that represent less than one individual of some species, then the practical interpretation that uniform persistence predicts coexistence may not be valid. An alternative approach is to seek asymptotic lower bounds on the populations or densities in the model, via comparison with simpler equations whose dynamics are better known. If such bounds can be obtained and approximately computed, then the prediction ofpersistence can be made practical rather than merely theoretical. This paper describes how practical persistence can be established for some classes of reaction–diffusion models for interacting populations. Somewhat surprisingly, themodels need not be autonomous or have any specific monotonicity properties.(Received May 17 1994)
In this article we study a finite-difference system, which is a discrete version of a class of nonlinear reaction-diffusion systems with time delays. Existence-comparison and uniqueness theorem are first established for the discretized problem by the method of upper and lower solutions. A monotone iterative scheme is also developed for the solution of the finite-difference system. Under a convergence acceleration scheme, it is shown that the monotone sequences converge quadratically to the solution of the finite-difference system. At last, numerical results on some model problems are demonstrated to substantiate our theorems. © 1995 John Wiley & Sons, Inc.
The classical two-species competition system is modified to include coefficients which are time-periodic with the same period. We show first that all (nonnegative) solutions converge to a periodic one, having the same period, thus excluding subharmonics. The global structure of the set of all periodic solutions is then investigated. This is accomplished by developing a geometric theory of the discrete dynamical system defined by the iterates of the period map T. It turns out, in particular, that periodic solutions appear which have no counterpart in the corresponding time-averaged system: thus oscillations in the environment may cause the two species to coexist in an oscillatory regime even if the corresponding averaged system would force either of the two species to extinction.
This paper considers a reaction-diffusion system that models the situation in which a predator feeds on two prey species. The existence of a positive steady-state solution as well as the existence and stability of various semitrivial steady-state solutions is obtained in terms of the natural growth rates of the three species. The present paper also gives some explicit information on the limiting behavior of the time-dependent solution to these steady states.
On the nonlinear differential equation y(t) = [A — By t — τ ]y(t), in Contributions to the Theory of Nonlinear Oscillations
- Kakutani