Aggregation and emergence in systems of ordinary differential equations

The aim of this article is to present aggregation methods for a system of ordinary differential equations (ODE's) involving two time scales. The system of ODE's is composed of the sum of fast parts and a perturbation. The fast dynamics are assumed to be conservative. The corresponding first integrals define a few global variables. Aggregation corresponds to the reduction of the dimension of the dynamical system which is replaced by an aggregated system governing the global variables at the slow time scale. The centre manifold theorem is used in order to get the slow reduced model as a Taylor expansion of a small parameter. We particularly look for the conditions necessary to get emerging properties in the aggregated model with respect to the nonaggregated one. We define two different types of emergences, functional and dynamical. Functional emergence corresponds to different functions for the two dynamics, aggregated and nonaggregated. Dynamical emergence means that both dynamics are qualitatively different. We also present averaging methods for aggregation when the fast system converges towards a stable limit cycle.     

Dynamics of a Predator-prey Model with Delay and Fear Effect

Recent manipulations on vertebrates showed that the fear of preda- tors, caused by prey after they perceived predation risk, could reduce the prey''s reproduction greatly. And it''s known that predator-prey systems with fear ef- fect exhibit very rich dynamics. On the other hand, incorporating the time delay into predator-prey models could also induce instability and oscillations via Hopf bifurcation. In this paper, we are interested in studying the com- bined effects of the fear effect and time delay on the dynamics of the classic Lotka-Volterra predator-prey model. It''s shown that the time delay can cause the stable equilibrium to become unstable, while the fear effect has a stabi- lizing effect on the equilibrium. In particular, the model loses stability when the delay varies and then regains its stability when the fear effect is stronger. At last, by using the normal form theory and center manifold argument, we derive explicit formulas which determine the stability and direction of periodic solutions bifurcating from Hopf bifurcation. Numerical simulations are carried to explain the mathematical conclusions.     

Predator-prey models in heterogeneous environment: Emergence of functional response

In this work, we are interested in prey-predator models. More precisely, we study the spatial heterogeneity effects on the amount of prey eaten per predator per unit time, when different time scales occur. This amount and its relation with the amount of predators produced via the predation are interesting from an ecological point of view. Indeed, the knowledge of these quantities permits us to quantify the transfer of the biomass in the food chain. Our aim is to show how the spatial heterogeneity acts on these amounts. We consider prey-predator systems in a multi-patch environment. We show that density dependent migrations make emerge new models on the total population level and we exhibit some examples. Furthermore, we show that the aggregation method is a good tool for describing the mechanisms hidden behind complex models.     

Expectation-Stock Dynamics in Multi-Agent Fisheries

In this paper we consider a game-theoretic dynamic model describing the exploitation of a renewable resource. Our model is based on a Cournot oligopoly game where n profit-maximizing players harvest fish and sell their catch on m markets. We assume that the players do not know the law governing the reproduction of the resource. Instead they use an adaptive updating scheme to forecast the future fish stock. We analyze the resulting dynamical system which describes how the fish population and the forecasts (expectations) of the players evolve over time. We provide results on the existence and local stability of steady states. We consider the set of initial conditions which give non-negative trajectories converging to an equilibrium and illustrate how this set can be characterized. We show how such sets may change as some structural parameters of our model are varied and how these changes can be explained. This paper extends existing results in the literature by showing that they also hold in our two-dimensional framework. Moreover, by using analytical and numerical methods, we provide some new results on global dynamics which show that such sets of initial conditions can have complicated topological structures, a situation which may be particularly troublesome for policymakers.     

Hawk-dove game and competition dynamics

In this article, we consider two populations subdivided into two categories of individuals (hawks and doves). Individuals fight to have access to a resource necessary for their growth. Conflicts occur between hawks of the same population and hawks of different populations. The aim of this work is to investigate the long term effects of these conflicts on coexistence and stability of the community of the two populations. This model involves four variables corresponding to the two tactics of individuals of the two populations. The model is composed of two parts, a fast part describing the encounters and fights, and the slow part describing the long term effects of encounters on the growth of the populations. We use aggregation methods allowing us to reduce this model into a system of two ODEs for the total densities of the two populations. This is found to be a classical Lotka-Volterra competition model. We study the effects of the different fast equilibrium proportions of hawks and doves in both populations on the global coexistence and the mutual exclusion of the two populations. We show that in some cases, mixed hawk and dove populations coexist. Aggressive populations of hawks exclude doves except in the case of interpopulation costs being smaller than intrapopulation ones.     

Competitive Lotka-Volterra population dynamics with jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the pth moment with p>0 and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our n-dimensional model.     

Optimal Harvesting and Stability for a Predator-prey System with Stage Structure

The dynamics of a predator-prey system, where prey population has two stages, an immature stage and a mature stage with harvesting, the growth of predator population is of Lotka-Volterra nature, are modelled by a system of retarded functional differential equations. We obtain conditions for global asymptotic stability of three nonnegative equilibria and a threshold of harvesting for the mature prey population. The effect of delay on the population at positive equilibrium and the optimal harvesting of the mature prey population are also considered.     

Dynamics of a reaction–diffusion–ODE system with quiescence

We consider a reaction–diffusion–ODE quiescent model in which the species can switch between mobile and immobile categories. We assume that the population inhabits a bounded region and study how its dynamics depend on the parameters describing switching rates and local population dynamics. Our results suggest that the transfer displays a stabilizing effect and inhibits the generation of spatial periodic solutions. A new method to obtain global stability and dissipative structure is also explored by constructing Lyapunov functionals to overcome the loss of compactness.     

From behavioural to population level: Growth and competition

The aim of this work is to build models of population dynamics for growth and competition interaction by starting with detailed models at the individual level. At the individual level, we start with detailed models where the growth is described by linear terms. By considering individual interferences and by using aggregation methods, we show that the population level, different growth equation can emerge. We present an example of the emergence of logistic growth and an example of the emergence of logistic growth with Allee effect. Furthermore, in the case of two populations, we show that individual interferences can lead at the population level, to a model which has the same qualitative dynamics behaviour as the Lotka-Volterra competition model. Finally, we show that our model brings to light the effects of spatial heterogeneity on competition models. First, we find the stabilizing effects but also we show that destabilizing effects can occur.     

A class of two-sex branching processes with reproduction phase in a random environment

We model the demographic dynamics of populations with sexual reproduction where the reproduction phase occurs in a non-predictable environment and we assume the immigration/out-migration of mating units in the population. We introduce a general class of two-sex branching processes where, in each generation, the number of mating units which take part in the reproduction phase is randomly determined and the offspring probability distribution changes over time in a random environment. We provide several probabilistic results about the limit behaviour of populations whose dynamics is modelled by such a class of stochastic processes. In particular, we provide sufficient conditions for the almost sure extinction of the population or for its survival with a positive probability. As illustration, we include some simulated examples.     

Global asymptotic stability of a two species competitive system with stage structure and harvesting

Introduction' There have recently appeared in the literature several mathematical models of stagestructured population growth, i. e., models which take into account the faCt that individuals in a population may belong to one of two classes, the immatures and the matureslllZI.Cannibalism has been observed in a great variety of species, including a number of fish species.Cannibalism models of various types have also been investigatedI3"l. In these models, the ageto maturity is represented by a…     

Analysis of nonautonomous two species system in a polluted environment

In this paper, a two-species nonautonomous Lotka-Volterra model of population growth in a polluted environment is proposed. Global asymptotic behaviour of this model by constructing suitable bounded functions has been investigated. It is proved that each population for competition, predation and cooperation systems respectively is uniformly persistent (permanent) under appropriate conditions. Sufficient conditions are derived to confirm that if each of competition, predation and cooperation systems respectively admits a positive periodic solution, then it is globally asymptotically stable.     

A Theoretical Approach to Understanding Population Dynamics with Seasonal Developmental Durations

There is a growing body of biological investigations to understand impacts of seasonally changing environmental conditions on population dynamics in various research fields such as single population growth and disease transmission. On the other side, understanding the population dynamics subject to seasonally changing weather conditions plays a fundamental role in predicting the trends of population patterns and disease transmission risks under the scenarios of climate change. With the host–macroparasite interaction as a motivating example, we propose a synthesized approach for investigating the population dynamics subject to seasonal environmental variations from theoretical point of view, where the model development, basic reproduction ratio formulation and computation, and rigorous mathematical analysis are involved. The resultant model with periodic delay presents a novel term related to the rate of change of the developmental duration, bringing new challenges to dynamics analysis. By investigating a periodic semiflow on a suitably chosen phase space, the global dynamics of a threshold type is established: all solutions either go to zero when basic reproduction ratio is less than one, or stabilize at a positive periodic state when the reproduction ratio is greater than one. The synthesized approach developed here is applicable to broader contexts of investigating biological systems with seasonal developmental durations.     

Neimark–Sacker bifurcation analysis in an intraguild predation model with general functional responses

We analyse the dynamics of a discrete system coming from an intraguild food web model by using the average method. The intraguild predation model is formed by three populations corresponding to prey (P), mesopredator (MP) and superpredator (SP), where these last two populations are specialist. We give sufficient condition to guarantee the existence of a coexistence point at which the intraguild predation discrete model undergoes a Neimark–Sacker bifurcation independently of the functional responses that govern the interactions. We show numerical applications that consist in to assume that P has logistic growth and that the relation of MP–P is through a Holling type II functional response. Besides, we will consider that the interaction of MP–P is such that population MP has defense. The interaction of SP–P will be through a Holling functional response type III or IV. In particular, we give sufficient conditions to guarantee that the three species coexist. The techniques used to obtain the results can be applied to other models with different functional responses.     

Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structureAnalyse de perturbation singulière pour la réduction de la chimie complexe des mélanges gazeux avec structure entropique

In this Note, we investigate the reduction of complex chemistry in gaseous mixtures. We consider an arbitrarily complex network of reversible reactions, the equilibrium constant of which are compatible with thermodynamics, thus providing an entropic structure. We assume that a subset of the reactions is consituted of fast reactions and define a constant and linear projection onto the partial equilibrium manifold compatible with the entropy production. This reduction step is used for the study of a homogeneous reactor at constant density and internal energy where the temperature can encounter strong variations. We prove the global existence of a smooth solution and of an asymptotically stable equilibrium state for both the reduced system and the complete one. A global in time singular perturbation analysis proves that the reduced system on the partial equilibrium manifold approximates the full chemistry system. To cite this article: M. Massot, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 93–98.     

Finite‐dimensional asymptotic behavior in a thermosyphon including the Soret effect

We analyse the dynamics of a fluid transporting a soluble substance in the interior of a closed loop of arbitrary geometry and subjected to the action of gravity and natural convection. After obtaining the governing equations and analysing the well posedness of the system we prove the existence of a global attractor. Finally, using inertial manifold techniques, we obtain an explicit reduced system of ODE's that describes the asymptotic behaviour of the full system. Copyright © 1999 John Wiley & Sons, Ltd.     

Hyperbolicity, transversality and analytic first integrals

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.     

Stationary probability distributions of some lotka-volterra types of stochastic predation systems

This paper provides exact solutions to the stationary probability distributions in some stochastic predation systems. These are derived by solving the Fokker-Planck equations for:

(i) a generalized stochastic Lotka-Volterra predator-prey system, and

(ii) a generalised stochastic Lotka-Volterra food chain.

In all these systems the growth dynamics of all levels of species are subject to stochastic shocks. Since stationary probability distributions provide the most comprehensive characterization of a stochastic system in a steady state, system stability can be analysed accordingly     

Nesting inertial manifolds for reaction and diffusion equations with large diffusivity

We study the asymptotic behaviour in large diffusivity of inertial manifolds governing the long time dynamics of a semilinear evolution system of reaction and diffusion equations. A priori, we review both local and global dynamics of the system in scales of Banach spaces of Hilbert type and we prove the existence of a universal compact attractor for the equations. Extensions yield the existence of a family of nesting inertial manifolds dependent on the diffusion of the system of equations. It is introduced an upper semicontinuity notion in large diffusivity for inertial manifolds. The limit inertial manifold whose dimension is strictly less than those of the infinite dimensional system of semilinear evolution equations is obtained.