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 robustness analysis of biological population models with protection zone

Establishing protection zone has been widely used to preserve endangered species. We study the effect of protection zone in random environment by a stochastic model. Our results show that the establishment of nature reserves can protect endangered species effectively in good conditions. Under very bad conditions, though protection zone cannot protect species from extinction, it also can slow down the speed of extinction. Based on our results, we give some methods to strengthen the effect of protection zone and provide a method of dividing the protection zone. In addition, we introduce some examples and use several numerical simulations to enhance our main results.     

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.     

Survival analysis of a single-species population model with fluctuations and migrations between patches

We formulate a single-species population model with fluctuations and migrations between two different patches (the nature reserve and the natural environment) to describe the situation that endangered species often meets with. Firstly, we prove that there exists a unique global positive solution to a stochastic single-species model with any positive initial value. Then, sufficient conditions that guarantee extinction and persistence in the mean of solutions are derived as the main results. It turns out that, under moderate fluctuation conditions, migration rates play significant roles for the persistence of endangered species in nature reserve and the natural environment. We further provide the sustainable strategies for local managers to avoid the extinction of endangered species, and separately carry out illustrative examples and numerical simulations at the end of this contribution.     

Modeling forest core area with integer programming

Abundant and continuous old forest tend to be fragmented into isolated and small patches because of human harvest activities. Dispersive and isolated old forest patches cannot provide abundant interior habitat to wildlife, which is a fatal threat for specific plant communities and wildlife species. In this paper, an Integer Programming model for forest planning is designed to maximize the economical benefit of the forest and to guarantee a minimum area of interior old forest for wildlife habitat, the so-called core area satisfying minimum mature age requirements. The minimum core area constraints, to some degree, can help mitigate the negative impact of harvest activities to divide forest habitat into many small patches. The model is implemented in a commercial Integer Programming solver and it is applied to several hypothetical landscapes. The results show the possibility of incorporating a core area requirement into a forest planning model, and the possibility to obtain solutions within a reasonable computational time. Instances with up to 1600 management units have been solved in seconds to an optimality gap of 1% (0.1% in some cases).     

Fuel treatment planning: Fragmenting high fuel load areas while maintaining availability and connectivity of faunal habitat

Reducing the fuel load in fire-prone landscapes is aimed at mitigating the risk of catastrophic wildfires but there are ecological consequences. Maintaining habitat for fauna of both sufficient extent and connectivity while fragmenting areas of high fuel loads presents land managers with seemingly contrasting objectives. Faced with this dichotomy, we propose a Mixed Integer Programming (MIP) model that can optimally schedule fuel treatments to reduce fuel hazards by fragmenting high fuel load regions while considering critical ecological requirements over time and space. The model takes into account both the frequency of fire that vegetation can tolerate and the frequency of fire necessary for fire-dependent species. Our approach also ensures that suitable alternate habitat is available and accessible to fauna affected by a treated area. More importantly, to conserve fauna the model sets a minimum acceptable target for the connectivity of habitat at any time. These factors are all included in the formulation of a model that yields a multi-period spatially-explicit schedule for treatment planning. Our approach is then demonstrated in a series of computational experiments with hypothetical landscapes, a single vegetation type and a group of faunal species with the same habitat requirements. Our experiments show that it is possible to fragment areas of high fuel loads while ensuring sufficient connectivity of habitat over both space and time. Furthermore, it is demonstrated that the habitat connectivity constraint is more effective than neighbourhood habitat constraints. This is critical for the conservation of fauna and of special concern for vulnerable or endangered species.     

A BIOECONOMIC APPROACH TO THE FAUSTMANN–HARTMAN MODEL: ECOLOGICAL INTERACTIONS IN MANAGED FOREST

Abstract This paper develops a bioeconomic forestry model that makes it possible to take ecosystem services that are independent of the age structure of trees into account. We derive the Faustmann–Hartman optimal harvesting strategy as a special case. The bioeconomic model is then extended to account for the fact that forest harvesting decisions impact on other ecological resources, which provide benefits for the wider community. The paper focuses on impacts associated with disturbance caused by logging operations and habitat destruction due to tree removal. This enables us to explore the interactions between forest management and the dynamics of ecological resources. The optimal rotation rule is obtained as a variation on the traditional Faustmann–Hartman equation, where an additional term captures the potential benefits derived from the growth of the ecological resource valued at its shadow price. The steady‐state solutions to the problem and sensitivity to model parameter are identified using numerical analysis.     

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.     

Spatial Dynamics of a Nonlocal Dispersal Population Model in a Shifting Environment

This paper is concerned with the spatial dynamics of a nonlocal dispersal population model in a shifting environment where the favorable region is shrinking. It is shown that the species becomes extinct in the habitat if the speed of the shifting habitat edge \(c>c^*(\infty )\), while the species persists and spreads along the shifting habitat at an asymptotic speed \(c^*(\infty )\) if \(c<c^*(\infty )\), where \(c^*(\infty )\) is determined by the nonlocal dispersal kernel, diffusion rate and the maximum linearized growth rate. Moreover, we demonstrate that for any given speed of the shifting habitat edge, the model system admits a nondecreasing traveling wave with the wave speed at which the habitat is shifting, which indicates that the extinction wave phenomenon does happen in such a shifting environment.     

The protection zone of biological population

Many species are endangered, if we do not act fast, we are going to lose them forever. Establishing protection zone is widely used to protect endangered species. How is the effect of this method? What factors affect the effect of the protection zone? We study this topic by a new mathematical model. We examine the effect of the protection zone and conclude that the protection zone is effective for conservation of population resources and ecological environment, though in some cases the extinction cannot be eliminated. The dangerous region, the parameters domains and the typical bifurcation curves of stability of steady states for the considered system are determined. Our results provide theoretical evidence for the practical management of biological resources.     

Logging damage and injured tree mortality in tropical forest management

Using insights from the forest ecology literature, we analyze the effect of injured trees on stand composition and carbon stored in above‐ground biomass and the implications for forest management decisions. Results from a Faustmann model with data for a tropical forest on Kalimantan show that up to 50% of the basal area of the stand before harvest can consist of injured trees. Considering injured trees leads to an increase in the amount of carbon in above‐ground biomass of up to 165%. These effects are larger under reduced impact logging than under conventional logging. The effects on land expectation value and cutting cycle are relatively small. The results suggest that considering injured trees in models for tropical forest management is important for the correct assessment of the potential of financial programs to store carbon and conserve forest ecosystem services in managed tropical forests, such as reducing emissions from deforestation and forest degradation and payment for ecosystem services. Recommendations for Resource Managers

• Considering the role of injured trees is important for managing tropical forests
• These trees can cover up to 50% of basal area and contain more than 50% of the carbon stored in above‐ground biomass
• Reduced impact logging leads to a larger basal area of injured trees and more carbon stored in injured trees than conventional logging
• Injured trees play an important role when assessing the potential for carbon storage in the context of payment for forest ecosystem services.

     

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.     

SUSTAINABLE FOREST MANAGEMENT FOR OPTIMIZING MULTISPECIES WILDLIFE HABITAT: A COASTAL DOUGLAS-FIR EXAMPLE

Wildlife species viability optimization models are developed to convert a given set of initial forest conditions, through a combination of natural growth and management treatments, to a forest system which addresses the joint habitat needs of multispecies populations over time. A linear model of forest cover and wildlife populations is used to form a system of forest management control variables for wildlife habitat modification. The paper examines two objective functions coupled to this system for optimizing sustainable joint species viability. The first maximizes the product of periodic joint viabilities over all time periods, focusing management resources on long-term equilibria, with less emphasis on conversion strategy. The second iteratively maximizes the minimum periodic joint viability over all time periods. This focuses management resources on the most limiting time periods, typically the conversion phase periods. Both objective functions resulted in either point or cyclic equilibria, with cycle lengths equal to minimum forest treatment ages. A third objective, based on maximizing the minimum individual species periodic viability is used to examine single species emphasis. Examples are developed through a case study of 92 vertebrate species found in coastal Douglas-fir stands of northwestern California.     

Dynamic Analysis of an Impulsive Chemostat Model with Microbial Competition and Nonlinear Perturbation

In this paper, we propose an impulsive chemostat model with microbial competition and nonlinear perturbation. First, thresholds for the extinction of both microoganisms are given. Second, we investigate the persistence in mean and boundedness of the chemostat system by constructing Lyapunov function. Moreover, we obtain the sufficient condition for the existence of an ergodic stationary distribution of the system. At last, numerical simulations are presented, and the results show that the competition between two species tends to make one species disappear from their common habitat, especially when the competition is concentrated in a single resource.     

Spatially explicit ecological models: a spatial convolution approach

Spatial structure tends to have a stabilizing influence on predator–prey interactions in which the local model predicts extinction of the system. This result is well supported by laboratory observations of simple systems. Here, we use a spatially explicit version of the Nicholson–Bailey model having Moran–Ricker host reproduction to repeat and extend some of these results. Our model is a discrete spatial convolution model analogous to the integrodifference equations (IDEs) used by other authors. We show a spatial rescue effect which prevents extinction of the system by reducing the size (standard deviation) of the dispersal pdf. We also show that very favorable habitat (K=∞) and marginal habitat (K=1.0), when mixed randomly together in an explicit map, are highly stabilizing whereas either kind of habitat alone will cause extinction. The marginal habitat in this situation has host densities below parasite replacement level and thus constitutes a host refuge (although not a complete one) from the parasite. When a host–parasitoid model having spiral wave dynamics in two-dimensional space was extended to one- and three-dimensional space, we observed analogous dynamics, i.e., traveling waves of evasion and pursuit in one dimension and ‘spiral-like’ structures in a three-dimensional spatial volume. We illustrate an approach to analysis of spatial convolution models via the frequency response of the system transfer function. In spatial convolution format, local interaction and dispersal are conveniently isolated from one another, and this allows us to vary these components independently and thus to study their effects on the dynamics of the total system. We show two examples of nonrandom dispersal pdf’s – a bimodal form representing two dispersal types in the population and a ‘ripple’ pdf representing a repulsive process.     

Spatial Dynamics of a Lattice Lotka-Volterra Competition Model with a Shifting Habitat

In this paper, we concern with the spatial dynamics of the lattice Lotka-Volterra competition system in a shifting habitat. We study the impact of the environmental deterioration rate on the population density under the strong competition condition. Our results show that if the environment deteriorates rapidly, both species will become extinct. However, when the environmental degradation rate is not so fast, the species with slow diffusion will go extinct, while those with fast diffusion will survive. The extinction of species with slow diffusion can be divided into two situations: one is the extinction caused by environmental deterioration faster than its own diffusion speed, the other is the extinction caused by slow diffusion speed under the influence of strong competition.     

LONG-TERM DYNAMIC ANALYSIS OF ENDANGERED SPECIES WITH STAGE-STRUCTURE AND MIGRATIONS IN POLLUTED ENVIRONMENTS

We propose a stochastic stage-structured single-species model with migrations and hunting within a polluted environment, where the species is separated into two groups: the immature and the mature, which migrates from one patch to another with different migration rates. By constructing a Lyapunov function, together with stochastic analysis approach, the stochastic single-species model admits a unique global positive solution. We then utilize the comparison theorem of stochastic differential equations to investigate the extinction and persistence of solution to stochastic single-species model. The main results indicate that the species densities all depend on the intensities of random perturbations within both patches. As a consequence, we further provide several strategies for protecting endangered species within protected and unprotected patches.     

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.     

A HABITAT BASED MODEL FOR THE DISTRIBUTION OF FOREST INTERIOR NESTING BIRDS IN A FRAGMENTED LANDSCAPE

ABSTRACT. Increased awareness of the plight of many forest dwelling species has made necessary the development of methods for projecting the spatial distribution of these populations. This is particularly important for populations that currently occupy forest fragments and that are likely to be exposed to further disruption of their natural habitat. In this paper we develop a model for predicting the distribution of a bird population that evolved as forest interior dwellers. This model uses as its basis knowledge of the relationship between demographic characteristics of the population and the qualities of the habitat where individuals reside. We make the assumption that individuals will be naturally drawn to areas where they might expect greater reproductive success and repelled from areas where there is a high degree of intraspecific competition (high density). We apply the model to the ovenbird population in a large region of the Midwest. We use the model to examine the relative extent to which the surplus production from two major source areas supports extensive sink populations. The basic diffusion model parameterized by county forest cover data projects a population distribution which compares favorably with the results from the breeding bird count.