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.     

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.     

Fast game dynamics coupled to slow population dynamics: A single population with hawk-dove strategies

We study a model of a population subdivided into two subpopulations corresponding to hawk and dove tactics. It is assumed that the hawk and dove individuals compete for a resource every Day, I.e., at a fast time scale. This fast part of the model is coupled to a slow part which describes the growth of the subpopulations and the long term effects of the encounters between the individuals which must fight to have an access to the resource. We aggregate the model into a single equation for the total population. It is shown that in the case of a constant game matrix, the total population grows according to a logistic curve whose τ and K parameters are related to the coefficients of the hawk-dove game matrix. Our result shows that high equilibrium density populations are mainly doves, whereas low equilibrium density populations are mainly hawks. We also study the case of a density dependent game matrix for which the gain is linearly decreasing with the total density.     

SIMULATIONS TO INVESTIGATE ANIMAL MOVEMENT EFFECTS ON POPULATION DYNAMICS

Abstract Movement of organisms will influence their ecology and demographics. Animal movements are often characterized by path structures with directional persistence. The extent to which this impacts on population dynamics is investigated in this paper using theoretical simulations. The effects of different movement strategies on variations in visits to individual patches across a forage area are discussed. Variations in resource biomass across patches are shown to persist after long simulations. Consumer resource equations with movement are used to show how heterogeneity in resources caused by these variations can affect global population dynamics. With directional movements (random or persistent), dynamics change from limit cycles to stable equilibrium solutions. It is suggested that this effect has the potential to increase survival because perturbations from unforeseen factors (like drought) are less likely to crash populations.     

Population model with diffusion and supplementary forest resource in a two-patch habitat

A mathematical model is proposed to study the role of supplementary self-renewable resource on species population in a two-patch habitat. It is assumed that the density of forest resource biomass is governed by the logistic equation in both the regions but with the different intrinsic growth rate but the same carrying capacity in the entire habitat. It is further assumed that the densities of species population is also governed by the generalized logistic equations in both the regions but with different growth rates and carrying capacities. It is shown that the steady state solutions are positive, monotonic and continuous under both reservoir and no-flux boundary conditions. The linear and non-linear asymptotic stability conditions of non-uniform steady state are compared with the case of the model with and without diffusion in a homogeneous habitat.     

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.     

Synergetic Approach and modeling of fish population dynamics

In the last few years it has become increasingly obvious that one of the obstacles in the way of constructing good simulation models of the global ocean ecosystem is a poor understanding of the general principles of marine ecosystems processes. A great number of factors and relationships acting in the marine environment, in combination with the random character of change in many of them, call for the development of new approaches in modeling. In this paper a synergetic approach is proposed. A new paradigm for this approach is discussed. As an example a population with logistic natural growth under different conditions of exploitation is considered. It is shown that the simplest mechanism, that principally changes the behavior of a population in a fluctuating environment, includes fishing and migration. This mechanism explains catastrophic changes in population abundance in cases when no one factor may be seen as exclusive. It is shown that the characteristic level of population number does not correspond to the average balance between input (migration), output (fishing) and the growth of the population. The environment variability leads to stabilization far from equilibrium. This totally conforms to one of the fundamental results in Synergetics which assert that nonequilibrium in the presence of fluctuations may serve as a source of new order.     

Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector

A delay ordinary deterministic differential equation model for the population dynamics of the malaria vector is rigorously analysed subject to two forms of the vector birth rate function: the Verhulst-Pearl logistic growth function and the Maynard-Smith-Slatkin function. It is shown that, for any birth rate function satisfying some assumptions, the trivial equilibrium of the model is globally-asymptotically stable if the associated vectorial reproduction number is less than unity. Further, the model has a non-trivial equilibrium which is locally-asymptotically stable under a certain condition. The non-trivial equilibrium bifurcates into a limit cycle via a Hopf bifurcation. It is shown, by numerical simulations, that the amplitude of oscillating solutions increases with increasing maturation delay. Numerical simulations suggest that the Maynard-Smith-Slatkin function is more suitable for modelling the vector population dynamics than the Verhulst-Pearl logistic growth model, since the former is associated with increased sustained oscillations, which (in our view) is a desirable ecological feature, since it guarantees the persistence of the vector in the ecosystem.     

Stochastic stable population growth

Classical demographic theory purports that the age structure of a population eventually stabilizes. Although the population may continue to grow, once equilibrium is reached, the proportions of people in different age categories do not change. Stochastic analogues can be proven if vital rates fluctuate according to a stationary stochastic process. The action of random matrix products on random vectors is studied. This permits the application of Hilbert's projective metric and leads to considerable simplification of the ergodic and central limit theory of population growth. Appropriate theorems and their proofs are presented.     

MODELING THE POPULATION DYNAMICS OF A SUBTROPICAL UNGULATE IN A VARIABLE ENVIRONMENT: RAIN,COLD AND PREDATORS

ABSTRACT. A structured population model was developed for a large ungulate, the kudu (Tragelaphus strepsiceros). From a ten-year study in South Africa's Kruger National Park, relationships were established between annual survival rates of particular age classes and resource availability indexed by the ratio between annual rainfall and population biomass density. The projected population dynamics resembled that from a simple logistic model, but with the convexity of density dependence and intrinsic growth rate dependent upon assumptions about how age-specific mortality changed at low density levels. Moreover, rather than being a preset constant, the effective carrying capacity K wasa dynamic variable dependent upon rainfall. The model closely replicated the observed dynamicsof the kudu population over the study period, but failed to predict the observed kudu density at the start of the study from prior rainfall alone. Episodic cold weather extremeswere identified ashaving an additional influence on kudu dynamics. The model was also unsuccessful in predicting the changesin kudu abundance that occurred in Kruger Park subsequent to the study. Here changes in predation perhaps due to predator switching were a possible influence. These additional factorsinfluencing population dynamicswould not have been recognized without first establishing the effects of changing resource availability in response to rainfall fluctu-ationsbetween years. The elaborated model incorporating the effects of resource supply as influenced by rainfall, density dependence, background predation pressure and episodic severe weather hasbroader reliability than simpler modelsfor conservation applications, while still having a firm empirical foundation.     

The stability of nonlinear age-dependent population equation

A kind of size-dependent age-structured single species population equation with a random gestation period is discussed. A generalized population size E, called “the newborn equivalent quantity” is defined. The stability of a positive equilibrium is studied when the control function is chosen to be E. It is proved that if E is unfavorable to both survival and reproduction, the unique positive equilibrium is globally asymptotically stable.     

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.     

Modeling the survival of a resource-dependent population: Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors

In this paper, a nonlinear mathematical model is proposed and analyzed to study the survival of a resource-dependent population. It is assumed that this population and its resource are affected simultaneously by a toxicant (pollutant) emitted into the environment from external sources as well as formed by precursors of this population. It is shown that as the cumulative rates of emission and formation of the toxicant into the environment increase, the densities of population and its resource settle down to lower equilibria than their initial carrying capacities, and their magnitudes decrease as rates of emission and formation of the toxicant increase. On comparing different cases, it is noted that when population is not affected directly by the toxicant but only its resource is affected, the possibility of its survival is greater than the case when both are affected simultaneously. But for large emission rate of toxicant, the affected resource may be driven to extinction under certain conditions and the population which wholly depends on it may not survive for long even if it is not affected directly by the toxicant.     

Technological change,population dynamics,and natural resource depletion

In this paper, we integrate fertility and educational choices into a scale-invariant model of directed technological change with non-renewable natural resources, in order to reveal the interaction between population dynamics, technological change, and natural resource depletion. In line with empirical regularities, skill-biased technological change induces a decline in population growth and a transitory increase in the depletion rate of natural resources. In the long-run, the depletion rate also declines in the skill intensity. A decline in population growth is harmful for long-run productivity growth, if R&D is subject to diminishing technological opportunities. The effectiveness of economic policies aimed at sustained economic growth thus hinges on its impact on long-run population growth given the sign of intertemporal spillovers in R&D with respect to existing technological knowledge. We demonstrate that an increase in relative research productivities or an education subsidy enhances long-run growth, if R&D is subject to diminishing technological opportunities, while an increase in the teacher–student ratio is preferable in terms of positive intertemporal knowledge spillovers.     

A discrete two-stage population model: continuous versus seasonal reproduction

A discrete two-stage model which describes the dynamics of a population where juveniles and adults compete for different resources is developed. A motivating example is the green tree frog (Hyla cinerea) where tadpoles and adult frogs feed on separate resources. First, continuous breeding is assumed and the asymptotic behavior of the resulting autonomous model is fully analyzed. It is shown that the unique interior equilibrium is globally asymptotically stable when the inherent net reproductive number is greater than one. However, when the inherent net reproductive number is less than one, the population becomes extinct. Then a seasonal breeding described by a periodic birth rate with period 2 is assumed. It is proved that for this nonautonomous model a period two solution is globally asymptotically stable when the inherent net reproductive number is greater than one and when the inherent net reproductive number is less than one the population becomes extinct. Finally, the advantage (in terms of maximizing the number of juveniles and adults in the population over a fixed time period) of having a seasonal breeding is studied by comparing the average of the juvenile and adult numbers of the periodic solution for the nonautonomous model to the equilibrium solution of the autonomous model. Our results indicate that for high birth rates the equilibrium of the autonomous model is higher than the average of the two cycle solution. Therefore, all other factors being equal, seasonal breeding appears to be deleterious to populations with high birth rates. However, for low birth rates seasonal breeding can be beneficial. It is also shown that for a range of birth rates the nonautnomous model is persistent while the solution to the autonomous model goes to extinction.     

Hierarchical Discrete Models of Competition for a Resource

A coupled pair of first order nonlinear discrete hierarchical age-structured models are applied to study two modes of intraspecific competitions; scramble and contest. The study focuses on several comparisons of the dynamical outcomes of the two competitions. For a constant resource, it is shown, using analytical and numerical approaches, that solutions of the contest model monotonically equilibrate, while solutions of the scramble model oscillate and become chaotic. It is also shown that the inherent net reproductive number of each population affects the comparison of equilibrium points in the two populations. By considering cases on the resource and model parameters, the local as well as the global stability of nontrivial equilibrium points are studied. The impact of a contest and a scramble consumer on a time dependent resource is considered numerically.     

Population Growth and Nash Equilibria Under Viability Constraints in the Commons

Population growth modifies the optimal equilibrium between a stationary population and its resource, producing instead a line of equilibria, characterized by fluctuating population size, resource quantity, harvest per head, and birth rates. The Pontryagin procedure allows the analytical expression of the Nash equilibria for two populations sharing a common resource and capable of growth. An alternative procedure, which avoids solving differential equations and inherently includes state constraints, involves building the capture-viability kernel of an auxiliary system. For two populations, all Nash equilibria under state constraints are obtained as the intersection of the boundaries of two capture-viability kernels. The two methods, Pontryagin and viability, yield concordant results. Viability is more flexible and avoids solving differential equations for each initial condition.     

Asymptotic Behavior on a Competition Model Arising from an Unstirred Chemostat

This paper is concerned with the unstirred chemostat model with two-species and one non-reproducingresource.The global attractivity of the positive steady-state solutions of the original system is established.Moreover,the effects of the growth rate on the unique positive equilibrium of the single population model arestudied.     

Chaos and order in stateless societies: Intercommunity exchange as a factor impacting the population dynamical patterns

The once abstract notions of dynamical chaos now appear naturally in various systems [Kaplan D, Glass L. Understanding nonlinear dynamics. New York: Springer; 1995]. As a result, future trajectories of the systems may be difficult to predict. In this paper, we demonstrate the appearance of chaotic dynamics in model human communities, which consist of producers of agricultural product and producers of agricultural equipment. In the case of a solitary community, the horizon of predictability of the human population dynamics is shown to be dependent on both intrinsic instability of the dynamics and the chaotic attractor sizes. Since a separate community is usually a part of a larger commonality, we study the dynamics of social systems consisting of two interacting communities. We show that intercommunity barter can lead to stabilization of the dynamics in one of the communities, which implies persistence of stable equilibrium under changes of the maximum value of the human population growth rate. However, in the neighboring community, the equilibrium turns into a stable limit cycle as the maximum value of the human population growth rate increases. Following an increase in the maximum value of the human population growth rate leads to period-doubling bifurcations resulting in chaotic dynamics. The horizon of predictability of the chaotic oscillations is found to be limited by 5 years. We demonstrate that the intercommunity interaction can lead to the appearance of long-period harmonics in the chaotic time series. The period of the harmonics is of order 100 and 1000 years. Hence the long-period changes in the population size may be considered as an intrinsic feature of the human population dynamics.     

Approximating the ideal free distribution via reaction-diffusion-advection equations

We consider reaction-diffusion-advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness, where fitness is defined as a logistic local population growth rate. We show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics. The modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution. The analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability. The models are quasilinear parabolic equations with nonlinear boundary conditions.