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.     

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 effect of time delay on the conservation of forestry biomass

In this paper, we have studied the effect of time delay on conservation of forestry biomass by proposing a non-linear mathematical model. In the modeling process, it is assumed that the density of forestry biomass depletes due to the presence of human population and it is being conserved by applying some technological efforts. The analysis of model shows that the density of forestry biomass may be conserved if the technological effort is applied within the appropriate time. A longer delay in applying technological effort for its conservation destabilizes the system. The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the mathematical results.     

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 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.     

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.     

USING RESERVES TO PROTECT FISH AND WILDLIFE SIMPLIFIED MODELING APPROACHES

ABSTRACT. This paper investigates theoretically to what extent a nature reserve may protect a uniformly distributed population of fish or wildlife against negative effects of harvesting. Two objectives of this protection are considered: avoidance of population extinction and maintenance of population, at or above a given precautionary population level. The pre‐reserve population is assumed to follow the logistic growth law and two models for post‐reserve population dynamics are formulated and discussed. For Model A by assumption the logistic growth law with a common carrying capacity is valid also for the post‐reserve population growth. In Model B, it is assumed that each sub‐population has its own carrying capacity proportionate to its distribution area. For both models, migration from the high‐density area to the low‐density area is proportional to the density difference. For both models there are two possible outcomes, either a unique globally stable equilibrium, or extinction. The latter may occur when the exploitation effort is above a threshold that is derived explicitly for both models. However, when the migration rate is less than the growth rate both models imply that the reserve can be chosen so that extinction cannot occur. For the opposite case, when migration is large compared to natural growth, a reserve as the only management tool cannot assure survival of the population, but the specific way it increases critical effort is discussed.     

Modeling the spread of an infectious disease with bacteria and carriers in the environment

An SIS model with immigration for the spread of an infectious disease with bacteria and carriers in the environment is proposed and analyzed. It is assumed that susceptibles get infected directly by infectives as well as by their contacts with bacteria discharged by infectives in the environment. The growth rate of density of bacteria is assumed to be proportional to the density of infectives and decreases naturally as well as by bacterial interactions with susceptibles and carriers. The carrier population density is considered to follow the logistic model and grows due to conducive human population density related factors. It is assumed further that the number of bacteria transported by carriers to susceptibles is proportional to densities of both bacteria and carriers. The model study shows that the spread of the infectious disease increases due to growth of bacteria and carriers in the environment and disease becomes more endemic due to immigration.     

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.     

PIGS,PARTIES, AND NOISE: HOW STOCHASTICITY IMPACTS THE ROBUSTNESS OF THE TSEMBAGA CULTURAL PRACTICES

Abstract We present a probabilistic perspective on sustainable resource usage. A mathematical model is introduced to describe the interplay between a population and its renewable resource base. The amount of effort a society chooses to exert in harvesting its resource is formalized in the model. Using an indigenous population of slash and burn farmers as a case study, we derive a system of stochastic differential equations from a system of ordinary differential equations introduced by another author. The cultural mechanisms that help to stabilize the population in the deterministic system actually decrease the expected survival time in the stochastic system.     

WILDLIFE RESERVES,POPULATIONS, AND HUNTING OUTCOME WITH SMART WILDLIFE

We consider a hunting area and a wildlife reserve and answer the question: How does clever migration decision affect the social optimal and the private optimal hunting levels and population stocks? We analyze this in a model allowing for two‐way migration between hunting and reserve areas, where the populations’ migration decisions depend on both hunting pressure and relative population densities. In the social optimum a pure stress effect on the behavior of smart wildlife exists. This implies that the population level in the wildlife reserve tends to increase and the population level in the hunting area and hunting levels tend to decrease. On the other hand, the effect on stock tends to reduce the population in the wildlife reserve and increase the population in the hunting area and thereby also increase hunting. In the case of the private optimum, open‐access is assumed and we find that the same qualitative results arise when comparing a situation with and without stress effects, but of course at a higher level of hunting. We also show that when net social benefits of hunting dominate the net social benefits of populations, wildlife reserves are optimally placed in areas of low carrying capacity and vice versa.     

MODELING AND ANALYSIS OF THE ACID RAIN FORMATION DUE TO PRECIPITATION AND ITS EFFECT ON PLANT SPECIES

Abstract In this paper, a nonlinear mathematical model is proposed and analyzed to study the formation of acid rain in the atmosphere because of precipitation and its effect on plant species. It is considered that acid‐forming gases such as SO₂ , NO₂ emitted from various sources combine with water droplets (moisture) during precipitation and form acid rain affecting plant species. It is assumed that the biomass density of plant species follows a logistic model and its growth rate decreases with increase in the concentration of acid rain. The model is analyzed by using stability theory of differential equations and numerical simulation. The model analysis shows that as the concentration of acid rain increases because of increase in the cumulative emission rates of acid forming gases, the biomass density of plant species decreases. It is noted that if the amount of acid formed becomes very large, the plant species may become extinct.     

MARINE RESERVES AS A MEASURE TO CONTROL BYCATCH PROBLEMS: THE IMPORTANCE OF MULTISPECIES INTERACTIONS

ABSTRACT. This paper explores the effects of using marine reserves as a measure to control bycatch that is of no commercial value, under different assumptions regarding the ecological interactions between targeted species and that taken as bycatch. Three cases are examined: (1) no ecological interactions between the two species, (2) targeted and bycatch species exist in a predator‐prey relationship and (3) species compete. Targeted species is assumed to consist of two sub‐populations that are discretely distributed in space, but linked through density dependent migration while bycatch species is assumed to consist of one uniformly distributed stock only. In each case the equilibrium stock levels of targeted and by‐catch species, effort and harvest are numerically calculated and compared, assuming pure open access and open access in combination with a reserve. It is of special interest to identify circumstances that allows for a win‐win situation, that is, both harvest of the targeted species and biomass of the bycatch species increase. It is shown that the ecological interactions between the two species influence the possibility of actually protecting the bycatch species through the use of a reserve, the possibility a win‐win situation, and the issue of what patch to close.     

OPTIMAL HARVESTING WITH DENSITY DEPENDENT RANDOM EFFECTS

We consider the effect of sudden large, randomly occurring density dependent disasters on the optimal harvest policy and optimal expected return for an exploited population. The population is assumed to grow logistically with disasters occurring on a time scale very short compared to the natural growth scale. The case of a density dependent disaster frequency is also treated. Stochastic dynamic programming is used in the optimization. For a set of realistic field data it is found that random effects typically have a significant effect on both optimal return and optimal effort levels. The effect of density dependence is far more pronounced for optimal return than for optimal effort levels.     

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.     

A resource-dependent competition model: Effects of toxicants emitted from external sources as well as formed by precursors of competing species

In this paper, a nonlinear mathematical model is proposed and analyzed to study the survival of resource-dependent competing species. It is assumed that competing species and its resource are affected simultaneously by a toxicant emitted into the environment from external sources as well as formed by precursors of competing species. Stabilities of all the equilibria are studied using the theory of differential equations and computer simulation. A condition which determines the persistence of the system is also obtained. It is concluded from the analysis that as the cumulative rates of emission and formation of toxicants into the environment increase, the densities of both competing species and its resource decrease. It is also concluded that the usual competitive outcomes for the resource biomass altered in the presence of precursors.     

能源消费对中国工业化进程制约作用的实证分析

本文运用经济计量方法,从不同角度全面考察了工业部门能源强度的变化趋势及工业化水平与能源需求之间的关联关系。结果表明,工业部门能源消费强度随中国工业化进程的推进呈下降趋势;中国工业化水平与能源消费之间存在长期均衡关系,这种长期均衡关系的短期调整幅度较大;中国工业化水平对能源需求的贡献份额较小,能源需求不会成为制约我国工业化进程中的瓶颈。从目前实际出发,对短期调整效应较弱的工业部门实行节能技术改造,走适合我国基本国情的、有助于资源节约型的新型工业化道路是目前及今后一个时期的必然选择。     

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.     

MODELING EFFECTS OF TWO INTERACTING POPULATIONS ON A RENEWABLE RESOURCE

Abstract In this paper, a general mathematical model is presented to study the effect of two populations on a resource biomass. The interaction between two populations is assumed to be competition, predation, or cooperation. These two populations may depend on the resource biomass partially, wholly, or they may predate on the resource. In each case, criteria for local stability, instability, and global stability are obtained. Numerical simulation is presented to illustrate the theoretical results obtained in each case. It is shown that the depletion of resource biomass is maximum in the case of cooperation and is minimum in the case of competition.     

Periodic solutions for an equation governing dynamics of a renewable resource subjected to Allee effects

In this article the minimum number of positive periodic solutions admitted by a non-autonomous scalar differential equation is estimated. This result is employed to find the minimum number of positive periodic solutions admitted by a model representing dynamics of a renewable resource that is subjected to Allee effects in a seasonally varying environment. The Allee effect refers to a decrease in population growth rate at low population densities. Leggett–Williams multiple fixed point theorem is used to establish the existence of positive periodic solutions.