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.     

Optimal control of swinging alliances in a parabolic competition model

A system of parabolic partial differential equations describes the interaction of three populations, modeling a dynamic competition/cooperation scenario. More precisely, two populations are always competing with each other, but the third population can switch the mode of alliance with the other two populations between cooperation and competition. The control is a function measuring the strength and nature of the alliance and the goal is to maximize the population with the swinging alliance while keeping the other two populations close to each other and minimizing the cost of the alliance action. Various scenarios are illustrated with numerical results.     

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.     

Modelling effects of industrialization,population and pollution on a renewable resource

In this paper, a mathematical model is proposed and analysed to study the simultaneous effect of industrialization, population and pollution on the depletion of a renewable resource. Criteria for local stability, global stability and instability are obtained. It is shown that if the densities of industrialization, population and pollution increase, then the density of the resource biomass decreases and it settles down at its equilibrium level whose magnitude is lower than its original carrying capacity. It is further noted that if these factors increase unabatedly, the resource biomass may be driven to extinction. Computer simulations are also performed to illustrate the results.     

Existence and uniqueness of solutions for a hierarchical system of two age-structured populations

We propose a class of new hierarchical model for the evolution of two interacting age-structured populations, which is a system of integro-partial differential equations with global feedback boundary conditions and may describe the interactions such as competition, cooperation and predator-prey relation. Based upon a group of natural conditions, the existence and uniqueness of solutions on infinite time interval are proved by means of fixed point and extension principle, and the continuous dependence of the solution on the initial age distribution is established. These results lay a sound basis for the investigation of stability, controllability and variable optimal control problems.     

MODELING EFFECTS OF PRIMARY AND SECONDARY TOXICANTS ON RENEWABLE RESOURCES

ABSTRACT. In this paper a nonlinear mathematical model to study effects of primary and secondary toxicants on the biomass of resources such as forestry, agricultural crops, etc., is proposed and analyzed. The primary toxicant is emitted into the environment with a constant prescribed rate by an external source and a part of which is transformed into a secondary toxicant, which is more toxic, both affecting the resource simultaneously. By using stability theory of differential equations, it is shown that the biomass density of resource attains an equilibrium level, the magnitude of which is smaller than its original (toxicant independent) carrying capacity and it decreases as the emission rate of primary toxicant increases. It is also shown that the decrease in biomass density of resource is more than the corresponding case of a single toxicant due to large transformation and uptake rates and high toxicity of secondary toxicant. It is pointed out that the resource may even become extinct if emission rate of primary toxicant and transformation rate of secondary toxicant are very large and their effects on resource are sufficiently harmful due to large uptake and high toxicity of secondary toxicant which is more toxic.     

A METAPHYSIOLOGICAL POPULATION MODEL OF STORAGE IN VARIABLE ENVIRONMENTS

ABSTRACT. We use mechanistic arguments to generalize a hierarchical metaphysiological approach developed by one of us to modeling biological populations (Getz, [1991, 1993]) and extend the approach to include a storage component in the population. We model the growth of single species and consumer-resource interactions, both with and without storage. Our approach unifies modeling storage across trophic levels and is much simpler and more efficient to implement numerically than individual based approaches or population approaches that include integral, delay, or partial differential equation components in the model. Using intake functions (i.e., functional responses) that include the effects of interference competition, we apply the model to a hypothetical herbivore feeding on a resource that fluctuates seasonally and demonstrate the importance of a flow from storage that buffers the population against periods when resources are scarce or absent. We also apply the model to a hypothetical plant population that is driven by fluctuating resources and demonstrate the importance of a translocation flow from storage at the end of a dormant season, corresponding to periods when resources are most scarce. Finally, we couple these two populations for the case where the herbivore feeds exclusively on non-storage biomass, and demonstrate how the population dynamics can be affected by the rates at which buffering and translocation flows transfer from storage to active tissue in the herbivore and plant populations. In particular, for certain buffering and translocation flow rates, 1-year unimodal, 2-year bimodal, and 2-year unimodal cycles can emerge in the same herbivore population.     

Interactional models for adults of two populations with maturation delays

This paper is concerned with interactional models for adults of two species delayed by their mature periods. The existence and local stability of equilibria are discussed thoroughly for competitive systems, cooperative systems and predator-prey systems, respectively. For systems with interaction of competition and cooperation, it is found that the two populations are uniformly persistent if the positive equilibrium is stable. For predator-prey interaction, however, some further conditions are needed to guarantee the persistence of the systems.     

Dynamics of a system of three interacting populations with Allee effects and stocking

A model of three interacting populations where two populations engage in competition and two populations are in predator–prey type interaction is proposed and analysed. One of the two competing populations is subject to Allee effects and is also a pest population. The other competing population is regarded as a control agent and is the host for the predator population. There is a constant level of the external control agents released into the interaction at each generation after parasitism. We provide asymptotic dynamics of the competition subsystem and prove that a Neimark–Sacker bifurcation occurs for the host–parasitoid subsystem when the unique interior steady state loses its stability. The three interacting populations are impossible to persist for all positive initial conditions. Sufficient conditions based on the initial population size of the population with Allee effects are derived for persistence of the three populations.     

Stability and limiting distributions of one and two species stochastic population models

Requirements for stability w.p.l. are developed for a variety of stochastic population models using the Itô formulation. These conditions are comparable, but different, from previously published research concerning requirements for stability in the second moment. Exact conditions for stability w.p.l. are presented for the two species Lotka-Volterra competition equations. These conditions are met by any deterministically stable set of such equations. As a byproduct of our study, log-normal type limiting distributions of populations are also explicitly obtained.     

一类竞争——合作生态模型平衡点的渐近稳定性

研究一类竞争-合作生态数学模型平衡点的渐近稳定性,得到了一组保证平衡点渐近稳定的充分条件.     

The more the better? Optimal degree of supply-chain cooperation between competitors

We consider the outsourcing strategy problem of two competing original equipment manufacturers (OEMs) whose products are each made up of two components. The OEMs have different specializations, and therefore the component that each firm can produce in-house is different. Each firm must decide whether to outsource the other component to the competing OEM or to a third-party supplier. Prior research has demonstrated that competitors can be better off cooperating as supply-chain partners; therefore, one might expect that, as long as the OEMs are not at a severe cost disadvantage, they should maximize their cooperation as supply-chain partners, especially when competition between products is strong. Interestingly, this study finds that more cooperation between competitors may actually be harmful. Under certain conditions, while one of the OEMs should outsource to the competing firm, the other should outsource to a third-party supplier, even when the third-party supplier is more expensive and the competition is intense.     

A model for the problem of the cooperation/competition between infinite continuous species

In this paper we consider a particular type of differential equation that we can consider as a simple model for the problem of the cooperation/competition of infinite species. In this model each of the species meets each of the other species with a degree of competition or cooperation and their arrangements affect the evolution of the species. A first result of the existence of a unique, local-in-time, solution is given.     

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.     

MODELING THE DEPLETION OF A RENEWABLE RESOURCE BY POPULATION AND INDUSTRIALIZATION: EFFECT OF TECHNOLOGY ON ITS CONSERVATION

Abstract In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of a renewable resource by population and industrialization with resource‐dependent migration. The effect of technology on resource conservation is also considered. In the modeling process, four variables are considered, namely, density of a renewable resource, population density, density of industrialization, and technological effort. Both the growth rate and carrying capacity of resource biomass, which follows logistic model, are assumed to be simultaneously depleted by densities of population and industrialization but it is conserved by technological effort. It is further assumed that densities of population and industrialization increase due to increase in the density of renewable resource. The growth rate of technological effort is assumed to be proportional to the difference of carrying capacity of resource biomass and its current density. The model is analyzed by using the stability theory of differential equations and computer simulation. The model analysis shows that the biomass density decreases due to increase in densities of population and industrialization. It decreases further as the resource‐dependent industrial migration increases. But the resource may never become extinct due to population and industrialization, if technological effort is applied appropriately for its conservation.     

Global asymptotic behavior in chemostat-type competition models with delay

The asymptotic behavior of solutions of a chemostat-type model in which two species compete for a limiting nutrient supplied at a constant rate is considered. The model incorporates a general nutrient uptake function and two distributed delays. The first delay models the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition and the second indicates that the growth of the species depends on the past concentration of the nutrient. Furthermore, it is assumed that there is interspecific competition between the two species as well as intraspecific competition within each species. Conditions for boundedness of solutions and existence of nonnegative equilibria are given. By constructing appropriate Liapunov-like functionals, some sufficient conditions for global attractivity of the positive equilibrium is obtained. The combined effects of the two different delays are studied. The main results of Freedman and Xu [H.I. Freedman, Y. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol. 31 (1993) 513–527] and Ruan and He [S. Ruan, X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math. 58 (1) (1998) 170–192] are improved and extended.     

Strategic interactions in traditional franchise systems: Are franchisors always better off?

The effects of price competition and advertising spillover on franchisees’ decision to cooperate and on franchisor’s contractual preferences are investigated. We show that the franchisees’ decision to cooperate or not depends on the type of franchise contracts. Under exclusive territory contracts, any mode of play between franchisees give the same profits to the franchisees and franchisor. Contracts that allow price competition and well targeted local advertising offer a good ground for horizontal cooperation, which may or may not benefit the franchisor depending on whether the prices are strategic substitutes or strategic complements. Contracts in which price competition is allowed and the burden of advertising decisions is totally transferred to the franchisor lead to cooperation between franchisees at the expense of the franchisor. Franchisees do not cooperate to the benefit of the franchisor if local advertising is predatory and price competition is not allowed in the contract, but franchisees are given the responsibility to undertake local advertising. Also, the franchisor endorses cooperation between franchisees when local advertising has a public good nature, but such a cooperation may never occur when the impact of local advertising on demand is significant. We finally show that while some contracts always dominate others, the choice of a franchise contract may also depend on local competition and/or the franchise goodwill.     

A model for processes combining competition with cooperation

Cooperative processes are usually treated separately from competitive processes. Such separation is often artificial, for there are a number of processes, at all levels, where cooperation intertwines with competition. A class of processes of this kind involving two component systems is described. The components are assumed to cooperate until they attain an optimum level, and to hinder each other's growth from then on. The model boils down to a system of non-linear equations which are solved in closed form for the most interesting case, the one where the process does not even get started unless there is cooperation.     

Impact of noise in a phytoplankton-zooplankton system

In this paper, we investigate the dynamics of a delayed toxic phytoplankton-two zooplankton system incorporating the effects of Levy noise and white noise. The value of this study lies in two aspects: Mathematically, we first prove the existence of a unique global positive solution of the system, and then we investigate the sufficient conditions that guarantee the stochastic extinction and persistence in the mean of each population. Ecologically, via numerical simulations, we find that the effect of white noise or Levy noise on the stochastic extinction and persistence of phytoplankton and zooplankton are similar, but the synergistic effects of the two noises on the stochastic extinction and persistence of these plankton are stronger than that of single noise. In addition, an increase in the toxin liberation rate or the intraspecific competition rate of zooplankton was found to be capable to increase the biomass of the phytoplankton but decrease the biomass of zooplankton. These results may help us to better understand the phytoplankton-zooplankton dynamics in the fluctuating environments.     

Dynamics of hierarchical models in discrete-time

We derive and analyze a general class of difference equation models for the dynamics of hierarchically organized populations. Different forms of intra-specific competition give rise to different types of nonlinearities. For our models, we prove that contest competition results asymptotically in only equilibrium dynamics. Scramble competition, on the other hand, can result in more complex asymptotic dynamics. We study both the case when the limiting resource is a constant and when it is dynamically modeled. We prove, in all cases, that the population persists if the inherent net reproductive number of the population is greater than one.