Mathematical models of restoration and control of a single species with Allee effect

According to the initial density of a single species with Allee effect and corresponding management strategy, three kinds of mathematical models are presented to describe the evolutionary process of the species by impulsive differential equations. When the initial density of the species is larger than economic injury level (EIL) (or economical threshold, ET), impulsive harvest control is considered in a finite time to decrease the population of the species. The feasibility of the impulsive harvest control in a finite time is given by the existence of solution of the model with initial and boundary value problem. When the initial density of the species is less than EIL (or ET), the model with state feedback control is formulated according to the state of the species. The existence and stability of periodic solution of the model with state feedback control are discussed. When the initial density of the species is less than the Allee threshold and the species tends to extinction, the model with impulsive release at fixed moments is presented to study the restoration of the species. The conditions for the feasibility of periodic restoration of the species are given. Finally, some discussions are given.     

A Kolmogorov-type competition model with multiple coexistence states and its applications to plant competition for sunlight

It is demonstrated that a Kolmogorov-type competition model featuring species allocation and gain functions can possess multiple coexistence states. Two examples are constructed: one in which the two competing species possess rectangular allocation functions but distinct gain functions, and the other in which one species has a rectangular allocation function, the second species has a bi-rectangular allocation function, and the two species share a common gain function. In both examples, it is shown that the species nullclines may intersect multiple times within the interior of the first quadrant, thus creating both locally stable and unstable equilibrium points. These results have important applications in the study of plant competition for sunlight, in which the allocation functions describe the vertical placement of leaves for two competing species, and the gain functions represent rates of photosynthesis performed by leaves at different heights when shaded by overlying leaves belonging to either species.     

THE VOLTERRA MODEL FOR THREE SPACIES PREDATOR-PREY SYSTEMS IN LOOPS

We consider the three species predator-prey model with the same intrinsic growth rates, where species 3 feeds on species 2, species 2 feeds on species 1, species 1 feeds on species 3. An open question raised by Nishan Krikorian is answered: We obtain the necessary and sufficient conditions for all the orbits to be unbounded. We also obtain the necessary and sufficient conditions for the positive equilibrium to be globally stable. It is shown that there exists a family of neutrally stable periodic orbits, in which we extend Darboux method to three-species models for the first time.     

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.     

Prime graphs and exponential composition of species

In this paper, we enumerate prime graphs with respect to the Cartesian multiplication of graphs. We use the unique factorization of a connected graph into the product of prime graphs given by Sabidussi to find explicit formulas for labeled and unlabeled prime graphs. In the case of species, we construct the exponential composition of species based on the arithmetic product of species of Maia and Méndez, and express the species of connected graphs as the exponential composition of the species of prime graphs.     

HARVEST PLANNING UNDER UNCERTAIN EXTINCTION TIME

ABSTRACT. Harvesting a targeted commercial species results in impacts on an endangered species which may have no direct market value but may need protection as mandated by environmental laws. A vigorous harvest path might drive the endangered species to extinction. This paper studies harvest planning when the endangered species is subject to an uncertain extinction time. It is shown that the optimal static harvest level under certainty can be used as an upper bound for the entire dynamic harvest time path under uncertainty. The paper also develops an explicit upper bound for the entire harvest path under uncertain extinction time of the endangered species. It is demonstrated that an increase in the hazard rate of the endangered species will shift down the upper bound for the entire harvest path, whereas an increase in the growth rate of the commercial species will shift up the upper bound for the entire harvest path.     

A mathematical model for chemical defense mechanism of two competing species

In this paper, a non-linear mathematical model is proposed and analyzed to study the phenomenon of a chemical defense mechanism involving two competing species, where each species produces a toxicant affecting the other. It is shown that if the emission rate coefficient of toxicant, produced by one species increases, the equilibrium density of the other species decreases and its magnitude is lower than its original carrying capacity. It is found that the usual principle of competitive exclusion (coexistence) in the absence of toxicant may change in the case under consideration. It is also observed that increases in the values of production rates of toxicants by the competing species and depletion rates of environmental toxicants due to its assimilation by the species has a destabilizing effect, and decrease in the washout rates of environmental toxicants has a destabilizing effect on the dynamics of the system. In the case of allelopathy, where only one species produces a toxicant affecting the other species, it is shown that the affected species is driven to extinction for large production rate of this toxicant.     

FROM HARVESTING TO NONHARVESTING UTILITY: AN OPTIMAL CONTROL APPROACH TO SPECIES CONSERVATION

The purpose of this paper is to retrace the evolution of mathematical models focused on relation and interaction between economic growth, sustainable development, and natural environment conservation. First, generic defensive expenditures are introduced into a common‐property harvesting model in order to favor the species growth. Second, a transition model comprising both harvesting and nonharvesting values of wildlife biological species emerges. The latter gives rise to a group of purely nonharvesting models where anthropic activities and economic growth may have positive or negative impact on the natural evolution of wildlife species. Several scholars have proved that optimal strategies that are relatively good for harvesting purposes are not simply “transferrable” to the context of conservation of wildlife biological species with no harvesting value. In addition, the existence of optimal policies for long‐term conservation of all biological species (with or without harvesting value) cannot be guaranteed without having relatively large species populations at the initial time. Therefore, all such strategies are incapable of enhancing the scarce populations of endangered species and, therefore, cannot save these species from eventual (local) extinction. As an alternative, policymakers may soon be compelled to design and implement short‐term defensive actions aimed at recovery and enhancement of endangered wildlife species.     

Competitive exclusion in a discrete juvenile–adult model with continuous and seasonal reproduction

We develop a general discrete juvenile–adult population model that describes two competing species. We consider species in which the juveniles only compete with other juveniles, and the adults only compete with other adults, i.e. juveniles and adults of either species do not compete. This is typical of amphibians where juveniles (tadpoles) live in water and adults (frogs) live on land. Assuming competition efficiencies of the two species are similar, we analyse the cases where reproduction is either continuous or seasonal. In both cases, we develop conditions on the invasion net reproductive numbers of the two species that will lead to competitive exclusion. We show using numerical simulations that coexistence and bistability are possible outcomes when competition efficiencies of the two species are different.     

The Stage-structured Predator-Prey System with Delay and Harvesting

In this article, we establish a mathematical model of two species with stage structure and the relation of predator-prey, obtain the necessary and sufficient condition for the permanence of two species and the extinction of one species or two species. We also obtain the optimal harvesting and the threshold of harvesting for the sustainable development and increasing delays can cause a bifurcation into periodic solutions. Finally, we give some numerical results and graphs.     

CHARACTERIZING SPECIES DISTRIBUTIONS BY PRODUCTIVITY AND MORTALITY RATES IN MULTISPECIES MODELS

Abstract The temporal change in species composition is one dimension that may be used to characterize ecosystems. Multispecies interactions and species abundance (or biomass) distributions can be analyzed by an N‐species Lotka–Volterra (LV) system of equations. We derive a statistical approximation to the stationary probability distribution of relative species biomass for a system of interacting species. The distribution is parameterized by the mean and variance of the LV species interaction coefficients and can exhibit a variety of shapes. Given this distribution, one can describe the state of the system, and subsequent perturbations within the system, in terms of shifts in the resulting stable distribution. Theoretical distributions with known properties were compared to distributions estimated from simulations and good agreement was found. Our analytical result could be used to develop “ecosystem indicators” for describing changes in the state of species within an ecosystem.     

Global stability for a model of competition in the chemostat with microbial inputs

We propose a model of competition of n species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n−1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

具有阶段结构和时滞的捕食系统

本文中 ,我们考虑具有阶段结构和两个时滞的两种群捕食系统 .对于时滞τ1+τ2 =0 ,我们得到了两个种群持续生存和一个种群或两个种群绝灭的充分必要条件 .当τ1+τ2 增加到正平衡点不稳定时 ,系统存在一个小振幅的周期解 .     

Sustainable management of an alpine national park: handling the two-edged effect of tourism

Attracting visitors to an alpine national park can open up additional sources of funding for species conservation. However, tourism also brings ecologically negative impacts to the park and, in particular, to endangered species. In this paper, we discuss the handling of this two-edged effect of nature-based tourism within the context of a national park’s management decision. We develop a stylized model which frames the interaction of a representative largely unknown species, its habitat, and park visitors in an alpine ecosystem. In applying this to the protection of a rock partridge population in the Hohe Tauern National Park (Austria), we illustrate that a combined visitor and species protection policy can maximize steady state net benefits from tourism and conservation, while ensuring that the endangered species reaches its conservation target in the long run. Thus, even for a small, largely unknown species such as the rock partridge, and not only for popular species like the golden eagle, it is possible to endogenously generate a conservation budget by attracting visitors.     

On the problem of modeling reflexive behavior in conflict by an example of biological communities

Under study is some model of a spatially distributed two-species population. The species interact according to the predator-prey pattern. Each species is distributed over two stations. The migration of individuals between the stations are purposeful and maximize the breeding coefficient for each species. In addition, each species pursues its reflexive strategy of behavior to determine the optimal migration flow.     

OPTIMAL MANAGEMENT OF INVASIVE SPECIES WITH DIFFERENT REPRODUCTION AND SURVIVAL STRATEGIES

Abstract In this paper, a numerical model is developed for analyzing the role of species life history and age structure for the optimal management of a commercial resident species that is exposed to an invasive species. It is shown that reproduction and mortality characteristics of both species ands age structure of the invader at the time of invasion are important for the costs of invasions when the invader and resident species compete for scarce resources. Commercially harvested species with low juvenile survival and high reproduction are found to be economically more robust against invasions. Species with these life‐history traits are also the most damaging as invaders. Properties of the harvesting cost function and the discount rate are shown to be of importance for the development of the invader population over time. Hence, it is possible to identify specific combinations of life‐history characteristics and economic conditions under which invasions cause particularly large economic damage.     

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.     

On a principle of predatory exclusion

Suppose two prey species have the same rate of reproduction, and they are subjected to predation. Then the species more susceptible to predation dies out. So that in effect the predator introduces competition between the prey species.     

Stochastic modeling of phytoplankton allelopathy

The study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research due to the global increase of harmful phytoplankton blooms. Ordinary differential equation models for two competing phytoplankton species, when one or both the species liberate toxic substances, are unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic formulation never predicts the sudden localized extinction of certain species. These obstacles of mathematical modeling can be overcome if we include stochastic variability in our modeling approach. In this investigation, we construct stochastic models of allelopathic interactions between two competing phytoplankton species as a continuous time Markov chain model as well as an Itô stochastic differential equation model. Approximate extinction probabilities for both species are obtained analytically for the continuous time Markov chain model. Analytical estimates are validated with the help of numerical simulations.