Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary

The structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined. In particular, a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary. Our existence results are established via the well-known sub-super solution method.     

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.     

Maximization of the total population in a reaction–diffusion model with logistic growth

This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction–diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang–bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (Nonlinear Anal Real World Appl 11(2):688–704, 2010). To this purpose, we compute the first and second variations of the total population. When the growth rate is not of bang–bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. When the first variation becomes zero, we prove that the second variation is positive. These results implies that the bang–bang property is essential for the maximization of total population.     

HUMAN FERTILITY DECISIONS AND COMMON PROPERTY RESOURCES: A DYNAMIC ANALYSIS

ABSTRACT. In rural areas of developing countries, parental decisions on number of offspring may be made on the basis of the role of children in harvesting local common property renewable resources. It has been argued that this may lead to a cycle of human over‐population and resource over‐exploitation. To investigate the plausibility of this argument, we present a discrete dynamic model with two state variables representing human population level N and resource stock level S. The model is similar to one given by Nerlove and Meyer but differs in several important respects. It is assumed that, in each over‐lapping generation of parents and children, parents decide how many children to have based on their resulting share of the local resource harvest and the costs associated with child‐rearing. Using simulation and analytical methods, the long term steady state population and resource stock levels for this dynamic noncooperative game are contrasted with the steady state when parental fertility decisions are made in a cooperative manner.     

On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains

We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.     

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.     

Asymptotic Profile of Species Migrating on a Growing Habitat

This paper deals with a diffusive logistic equation on one dimensional isotropically growing domain. The model equation on growing domains is first presented, and the comparison principle is then proved. The asymptotic behavior of temporal solutions to the reaction-diffusion problem is given by constructing upper and lower solutions. Our result shows that when the domain grows slowly, the species successfully spreads to the whole habitat and stabilizes at a positive steady state, while it dies out in the long run if the domain grows fast. Numerical simulations are also presented to illustrate the analytical result.     

Does movement toward better environments always benefit a population?

We study the effects of advection along environmental gradients on logistic reaction-diffusion models for population growth. The local population growth rate is assumed to be spatially inhomogeneous, and the advection is taken to be a multiple of the gradient of the local population growth rate. It is also assumed that the boundary acts as a reflecting barrier to the population. We show that the effects of such advection depend crucially on the shape of the habitat of the population: if the habitat is convex, the movement in the direction of the gradient of the growth rate is always beneficial to the population, while such advection could be harmful for certain non-convex habitats.     

Non-constant positive steady state of a diffusive Leslie-Gower type food web system

A three species food web comprising of two preys and one predator in an isolated homogeneous habitat is considered. The preys are assumed to grow logistically. The predator follows modified Leslie-Gower dynamics and feeds upon the prey species according to Holling Type II functional response. The local stability of the constant positive steady state of the corresponding temporal system and the spatio-temporal system are discussed. The existence and non-existence of non-constant positive steady states are investigated.     

Flow in a porous medium induced by torsional oscillation of a disk near its surface

Torsional oscillation of an infinite disk in a viscous liquid bounded by a porous medium fully saturated with the liquid has been discussed. It is assumed that the flow between the disk and the porous medium is governed by Navier-Stokes equation and that in the porous medium by Brinkman equation. Flows in the two regions are matched at the interface by assuming that the velocity and stress components are continuous at it. It is found that the depth of penetration of the flow in the porous medium is proportional to the square root of the permeability of the medium. The oscillation of the disk induces a steady radial-axial flow in both the regions in such a way that there is a steady axial flow of the fluid from the porous medium to the free flow region i.e. the fluid is expelled out from the porous medium. The steady flow in the porous medium increases with the increase of the permeability of the medium and with the decrease of the distance between the oscillating disk and porous surface.     

Estimating a Resource Selection Function With Line Transect Sampling

A resource selection probability function is a function that gives the probability that a resource unit (e.g., a plot of land) that is described by a set of habitat variables X1 to Xp will be used by an animal or group of animals in a certain period of time. The estimation of a resource selection function is usually based on the comparison of a sample of resource units used by an animal with a sample of the resource units that were available for use, with both samples being assumed to be effectively randomly selected from the relevant populations. In this paper the possibility of using a modified sampling scheme is examined, with the used units obtained by line transect sampling. A logistic regression type of model is proposed, with estimation by conditional maximum likelihood. A simulation study indicates that the proposed method should be useful in practice.     

A cannibalistic eco-epidemiological model with disease in predator population

In this present article, we propose and analyze a cannibalistic predator–prey model with disease in the predator population. We consider two important factors for the dynamics of predator population. The first one is governed through cannibalistic interaction, and the second one is governed through the disease in the predator population via cannibalism. The local stability analysis of the model system around the biologically feasible equilibria are investigated. We perform global dynamics of the model using Lyapunov functions. We analyze and compare the community structure of the system in terms of ecological and disease basic reproduction numbers. The existence of Hopf bifurcation around the interior steady state is investigated. We also derive the sufficient conditions for the permanence and impermanence of the system. The study reveals that the cannibalism acts as a self-regulatory mechanism and controls the disease transmission among the predators by stabilizing the predator–prey oscillations.     

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.     

The role of indirect prey-taxis and interference among predators in pattern formation

It is important to find biological factors that may lead to formation of patches in the distribution of species. We build a simple model describing a consumer/predator which, besides random dispersion, searches for food by moving toward the gradient of some chemical released by prey. This mechanism is referred as indirect prey-taxis. The predator's rate of consumption is assumed to drop due to interference among predators when too many of them encounter on some aggregate of prey. The latter is assumed to have a negligible motility with its density governed by an ODE. The interference among consumers is modeled by a modification of the Beddington's de Angelis functional response. Detailed bifurcation analysis and numerical simulations of an auxiliary system indicate that nonconstant steady states imitating patches of species occur in the model provided the predator density exceeds certain threshold and taxis strength is big enough.     

On the effects of migration and spatial heterogeneity on single and multiple species

We first investigate in a logistic model the effects of migration and spatial heterogeneity of the environment on the total population size at equilibrium of a single species. Our study shows that (i) the total population size is maximized at some intermediate migration rate, and hence is a non-monotone function of the migration rate; (ii) heterogeneity of the environment increases the population size. In the second part of this paper, these findings are applied to ecological invasions. For a two-species Lotka-Volterra competition model with migration, we show that (i) without migration, the invading species eliminates the resident species at every point of the habitat, whereas when migration is present, for certain ranges of migration rates the invader may be eliminated when it is rare; and (ii) without migration, the two species can coexist at every point of the habitat, whereas when migration is present, for some ranges of migration rates one of the species is extinguished for all initial conditions.     

Global dynamics of a competition model with non-local dispersal I: The shadow system

Equations with non-local dispersal have been widely used as models in biology. In this paper we focus on logistic models with non-local dispersal, for both single and two competing species. We show the global convergence of the unique positive steady state for the single equation and derive various properties of the positive steady state associated with the dispersal rate. We investigate the effects of dispersal rates and inter-specific competition coefficients in a shadow system for a two-species competition model and completely determine the global dynamics of the system. Our results illustrate that the effect of dispersal in spatially heterogeneous environments can be quite different from that in homogeneous environments.     

Bifurcations on hemispheres

Summary It is now well known that the number of parameters and symmetries of an equation affects the bifurcation structure of that equation. The bifurcation behavior of reaction-diffusion equations on certain domains with certain boundary conditions isnongeneric in the sense that the bifurcation of steady states in these equations is not what would be expected if one considered only the number of parameters in the equations and the type of symmetries of the equations. This point was made previously in work by Fujii, Mimura, and Nishiura [6] and Armbruster and Dangelmayr [1], who considered reaction-diffusion equations on an interval with Neumann boundary conditions.As was pointed out by Crawford et al. [5], the source of this nongenericity is that reaction-diffusion equations are invariant under translations and reflections of the domain and, depending on boundary conditions, may naturally and uniquely be extended to larger domains withlarger symmetry groups. These extra symmetries are the source of the nongenericity. In this paper we consider in detail the steady-state bifurcations of reaction-diffusion equations defined on the hemisphere with Neumann boundary conditions along the equator. Such equations have a naturalO(2)-symmetry but may be extended to the full sphere where the natural symmetry group isO(3). We also determine a large class of partial differential equations and domains where this kind of extension is possible for both Neumann and Dirichlet boundary conditions.     

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers

• Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs).
• These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states.
• The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts.
• Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.

     

Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor

A mathematical model of tumor cell population dynamics is considered. The tumor is assumed to consist of cells of two types: amenable and resistant to chemotherapeutic treatment. It is assumed that the growth of the cell populations of both types is governed by logistic equations. The effect of a chemotherapeutic drug on the tumor is specified by a therapy function. Two types of therapy functions are considered: a monotonically increasing function and a nonmonotone one with a threshold. In the former case, the effect of a drug on the tumor is stronger at a higher drug concentration. In the latter case, a threshold drug concentration exists above which the effect of the therapy reduces. The case when the total drug amount is subject to an integral constraint is also studied. A similar problem was previously studied in the case of a linear therapy function with no constraint imposed on the drug amount. By applying the Pontryagin maximum principle, necessary optimality conditions are found, which are used to draw important conclusions about the character of the optimal therapy strategy. The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.