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.     

A density-dependent model describing age-structured population dynamics using hawk–dove tactics

In this paper we deal with a nonlinear two-timescale discrete population model that couples age-structured demography with individual competition for resources. Individuals are divided into juvenile and adult classes, and demography is described by means of a density-dependent Leslie matrix. Adults compete to access resources; every time two adults meet, they choose either being aggressive (hawk) or non-aggressive (dove) to get the best pay-off. Individual encounters occur much more frequently than demographic events, what yields that the model takes the form of a two-timescale system. Approximate aggregation methods allow us to reduce the system while preserving at the same time crucial asymptotic information for the whole population. In this way, we are able to describe the total population size as function of individual aggressiveness level and environmental richness. Model analysis shows a general trend with species that look for richer environment having smaller proportions of hawk individuals with larger costs.     

On some variants of dynamical systems

A dynamical system motivated by discrete physics is studied. Fuzzy dynamical systems are used to study fuzzy discrete replicator dynamics of hawk–dove (HD) and prisoner's dilemma (PD) games. New solutions are obtained. Finally a preliminary study for fuzzy predator–prey model is studied and a new equilibrium is found.     

A model of an age-structured population in a multipatch environment

We propose a model of an age-structured population divided into N geographical patches. We distinguish two time scales, at the fast time scale we have the migration dynamics and at the slow time scale the demographic dynamics. The demographic process is described using the classical McKendrick-von Foerster model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process.Assuming that 0 is a simple strictly dominant eigenvalue for the migration matrix, we transform the model (an e.d.p. problem with N state variables) into a classical McKendrick-von Foerster model (scalar e.d.p. problem) for the global variable: total population density. We prove, under certain assumptions, that the semigroup associated to our problem has the property of positive asynchronous exponential growth and so we compare its asymptotic behaviour to that of the transformed scalar model. This type of study can be included in the so-called aggregation methods, where a large scale dynamical system is approximately described by a reduced system. Aggregation methods have been already developed for systems of ordinary differential equations and for discrete time models.An application of the results to the study of the dynamics of the Sole larvae is also provided.     

Mathematical study of a risk-structured two-group model for Chlamydia transmission dynamics

A new two-group deterministic model for Chlamydia trachomatis, which stratifies the entire population based on risk of acquiring or transmitting infection, is designed and analyzed to gain insight into its transmission dynamics. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) co-exists with one or more stable endemic equilibria when the associated reproduction number is less than unity. Unlike in some of the earlier modeling studies on Chlamydia transmission dynamics in a population, this study shows that the backward bifurcation phenomenon persists even if individuals who recovered from Chlamydia infection do not get re-infected. However, it is shown that the phenomenon can be removed if all the susceptible individuals are equally likely to acquire infection (i.e., for the case where the susceptible male and female populations are not stratified according to risk of acquiring infection). In such a case, the DFE of the resulting (reduced) model is globally-asymptotically stable when the associated reproduction number is less than unity and no re-infection of recovered individuals occurs. Thus, this study shows that stratifying the two-sex Chlamydia transmission model, presented in [1], according to the risk of acquiring or transmitting infection induces the phenomenon of backward bifurcation regardless of whether or not the re-infection of recovered individuals occurs.     

Modelling the depletion of forestry resources by population and population pressure augmented industrialization

In this paper, a nonlinear mathematical model is proposed and analysed to study the depletion of forestry resources caused by population and population pressure augmented industrialization. It is shown that the equilibrium density of resource biomass decreases as the equilibrium densities of population and industrialization increase. It is found that even if the growth of population (whether intrinsic or by migration) is only partially dependent on resource, still the resource biomass is doomed to extinction due to large population pressure augmented industrialization. It is noted that for sustained industrialization, control measures on its growth are required to maintain the ecological stability.     

Global Analysis of Predator–Prey System with Hawk and Dove Tactics

Functional response of the Holling type II is incorporated into a predator–prey model with predators using hawk‐dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle‐node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.     

On an N-person noncooperative Markov game with a metric state space

We consider a noncooperative N-person discounted Markov game with a metric state space, and define the total expected discounted gain. Under some conditions imposed on the objects in the game system, we prove that our game system has an equilibrium point and each player has his equilibrium strategy. Moreover in the case of a nondiscounted game, the total expected gain up to a finite time can be obtained, and we define the long-run expected average gain. Thus if we impose a further assumption for the objects besides the conditions in the case of the discounted game, then it is proved that the equilibrium point exists in the nondiscounted Markov game. The technique for proving the nondiscounted case is essentially to modify the objects of the game so that they become objects of a modified Markov game with a discounted factor which has an equilibrium point in addition to the equilibrium point of the discounted game.     

Baseline models of spatial population dynamics

To understand human population dynamics fully, before considering complex human agency it may be useful to construct baseline models to see where such agency may and may not be necessary. In fact, the dynamics of human populations may be amenable to mathematical modeling with relatively parsimonious mechanisms. We review some of the more prominent of such models, namely, the spatial Galton-Watson (GW) model, modifications of the GW model that add migration and immigration, and the Bolker-Pacala model, in which mortality (or birth rate) is affected by competition. We show that change in the distribution of population density over the last century for 12 American rural states may be captured by the simplest of the models, the spatial GW model.     

Modeling herd behavior in population systems

In this paper, we show that under suitable simple assumptions the classical two populations system may exhibit unexpected behaviors. Considering a more elaborated social model, in which the individuals of one population gather together in herds, while the other one shows a more individualistic behavior, we model the fact that interactions among the two occur mainly through the perimeter of the herd. We account for all types of populations’ interactions, symbiosis, competition and the predator–prey interactions. There is a situation in which competitive exclusion does not hold: the socialized herd behavior prevents the competing individualistic population from becoming extinct. For the predator–prey case, sustained limit cycles are possible, the existence of Hopf bifurcations representing a distinctive feature of this model compared with other classical predator–prey models. The system’s behavior is fully captured by just one suitably introduced new threshold parameter, defined in terms of the original model parameters.     

A model for an age-structured population with two time scales

In the modelisation of the dynamics of a sole population, an interesting issue is the influence of daily vertical migrations of the larvae on the whole dynamical process. As a first step towards getting some insight on that issue, we propose a model that describes the dynamics of an age-structured population living in an environment divided into N different spatial patches. We distinguish two time scales: at the fast time scale, we have migration dynamics and at the slow time scale, the demographic dynamics. The demographic process is described using the classical McKendrick model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that the migration process is conservative with respect to the total population and some additional technical assumptions, we proved in a previous work that the semigroup associated to our problem has the property of positive asynchronous exponential growth and that the characteristic elements of that asymptotic behaviour can be approximated by those of a scalar classical McKendrick model. In the present work, we develop the study of the nature of the convergence of the solutions of our problem to the solutions of the associated scalar one when the ratio between the time scales is ε (0 < ε ⪡ 1). The main result decomposes the action of the semigroup associated to our problem into three parts:

• 1.(1) the semigroup associated to a demographic scalar problem times the vector of the equilibrium distribution of the migration process;
• 2.(2) the semigroup associated to the transitory process which leads to the first part; and
• 3.(3) an operator, bounded in norm, of order ε.

     

Variational formulation of an age-physiology dependent population dynamics

We transform a deterministic age-physiological factor population dynamics problem into its variational form. The internal/external heterogeneity of a population profoundly affects its dynamics, therefore, apart from age a, a second independent variable, g, say, referred to as the physiological parameter of individuals will also be a basis for classification. Using the well-known Ostrogradski or Gauss formula, we prove the existence and uniqueness theorems for the classical weak solution of the model.     

A mathematical model of the dynamics of a population affected by harmful substances

A mathematical model is presented of the dynamics of a population with individuals subjected to the effects of pollutants entering with food. We assume that the product of interaction of ingested pollutants is harmful for the individuals and increases the rate of their death. We describe the equations of the model and study the properties of solutions, including the existence and stability of equilibria. The conditions are obtained for the population becoming extinct as well as the conditions which guarantee that the total population is maintained at a nonzero stationary level. Some results of simulation are presented.     

A bifurcation theorem for nonlinear matrix models of population dynamics

ABSTRACT

We prove a general theorem for nonlinear matrix models of the type used in structured population dynamics that describes the bifurcation that occurs when the extinction equilibrium destabilizes as a model parameter is varied. The existence of a bifurcating continuum of positive equilibria is established, and their local stability is related to the direction of bifurcation. Our theorem generalizes existing theorems found in the literature in two ways. First, it allows for a general appearance of the bifurcation parameter (existing theorems require the parameter to appear linearly). This significantly widens the applicability of the theorem to population models. Second, our theorem describes circumstances in which a backward bifurcation can produce stable positive equilibria (existing theorems allow for stability only when the bifurcation is forward). The signs of two diagnostic quantities determine the stability of the bifurcating equilibrium and the direction of bifurcation. We give examples that illustrate these features.     

On simple age-structured population models

This work addresses several aspects and extensions of the deterministic Leslie model, as a matrix-driven demographic evolution of an age-structured population. We first point out its duality with another matrix model, related to backward/forward in time ways of counting individuals. Then, in some special cases, we design explicitly both the eigenvalues and the offspring vector of the Leslie matrix in a consistent way. Finally, we show how embedding the dynamics in a space of larger dimension allows one to get various new results about the population. This includes access to the total lifetime asymptotic distribution and while including sterile and/or immortal individuals in the classical Leslie model, some insight into the trade-off between the different population species.     

COMPETITION AND COOPERATION IN A DYNAMICAL MODEL OF NATURAL RESOURCES

Abstract In this paper, we propose a model describing the commercial exploitation of a common renewable resource by a population of strategically interacting agents. Players can cooperate or compete; cooperators maximize the payoff of their group while defectors maximize their own profit. The partition of the players into two groups, defectors and cooperators, results from the players' choices, so it is not predetermined. This partition is decided as a Nash equilibrium of a static game. It is shown that different types of players can exist in an equilibrium; more precisely, depending on the parameter values such as resource stock, cost, and so on, there might be equilibria only with defectors, cooperators, or with a combination of cooperators and defectors. In any case the total harvest depends on the renewable resource stock, so it influences agents' positions. It is assumed that at each time period the agents harvest according to Nash equilibrium, which can be combined with a dynamic model describing the evolution of fish population. A complete analysis of the equilibria is presented and their stability is analysed. The effect of the different Nash equilibria on the stability of the fish stock, showing that full cooperation is the most stable case, is examined.     

Dynamic model and simulation analysis of the genetic impact of population harvesting

In this paper the impact of exploitation of a sexually reproducing population is investigated by means of a selection model. Our aim is to find genetic systems for which the fishing results in selection effect, provided that the growth and reproduction rates of individuals are genetically determined. To this end a complex dynamic model is presented, providing long-term predictions both on the size structure and on the genetic composition of the population. For a minimal nontrivial model, the two-locus two-allele case is considered, where the survival, transition and reproduction rates depend on size and genotype. For each size class and genotype the corresponding density is a state variable. The mating system is supposed to be totally panmictic and the gamete production is described in terms of the meiosis matrix.Based on the above model, an in silico analysis is carried out. The simulation results show that the long-term behaviour of the genetic structure can be characterized by a cyclic convergence, which means that the state sequences corresponding to different phases of the reproduction cycle tend to an asymptotic genotype distribution. For an illustration of the effect of exploitation on the genetic composition the “fishing effort” model is considered. If the totally homozygous genotype possesses the best phenotype, fishing does not seem to influence the genetic distribution in the long term. The same is true in case of heterozygote advantage. In some situations, however, fishing modifies the genetic distribution of the population. Meanwhile there is a significant change in the size of the harvested individuals. This result points out to the importance of the genotype-phenotype correspondence while building up fishing strategies.     

A semigroup approach to age-dependent population dynamics with time delay

We study the McKendrick type models of population dynamics with instantaneous time delay in the birth rate. The models involve first order partial differential equations with nonlocal and delayed boundary conditions. We show that a semigroup can be associated

to it and identify the infinistimal generator. Its spectral properties are analyzed yielding large time behaviour. An interesting result is that if the total population converges to an equilibrium it will converge to it in an oscillatory fashion. Further, we consider a logistic ara age-dependent model with delay. A nonlinear semigroup is constructed to describe the evolution of the population. Existence and uniqueness of the nonlinear equation are proved.     

Unit-sphere games

This paper introduces a class of games, called unit-sphere games, in which strategies are real vectors with unit 2-norms (or, on a unit-sphere). As a result, they should no longer be interpreted as probability distributions over actions, but rather be thought of as allocations of one unit of resource to actions and the payoff effect on each action is proportional to the square root of the amount of resource allocated to that action. The new definition generates a number of interesting consequences. We first characterize the sufficient and necessary condition under which a two-player unit-sphere game has a Nash equilibrium. The characterization reduces solving a unit-sphere game to finding all eigenvalues and eigenvectors of the product matrix of individual payoff matrices. For any unit-sphere game with non-negative payoff matrices, there always exists a unique Nash equilibrium; furthermore, the unique equilibrium is efficiently reachable via Cournot adjustment. In addition, we show that any equilibrium in positive unit-sphere games corresponds to approximate equilibria in the corresponding normal-form games. Analogous but weaker results are obtained in n-player unit-sphere games.     

Equilibrium path in oligopolistic market of nonrenewable resource

In this paper we study the oligopoly model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. We consider both the open-loop strategy and the closed-loop strategy of this dynamical differential game. For the case of linear cost function we have observed that the open-loop equilibrium and the self-feedback equilibrium satisfy the same equilibrium conditions, which can be described as a dynamical system. The analysis shows that the equilibrium path of the model is the stable orbit of this system, and this result leads to further studies of the properties of the total extraction and reserve and the individual ones of each producer. For the total extraction rate and reserve, some of the properties are similar to those of most oligopoly models with fixed unit production cost. For the individual behaviors, we have found out the solution expressions of the individual extraction rate and resource reserve and got the main result that the producer with larger initial stock has a larger but declining market share and the share of each producer converges toward the average one when time approaches to infinite.