Optimal harvesting policy for stochastic Logistic population model

Optimal impulsive harvesting on non-autonomous Beverton–Holt difference equations

Time-dependent probability density function for general stochastic logistic population model with harvesting effort

Optimal impulsive control in periodic ecosystem

We consider the optimal harvesting and planting control problem to maximize the expected total net benefits in the stochastic logistic population model. The variational inequality associated with this problem is given by the degenerate form of elliptic type with quadratic coefficients. Using the viscosity solutions technique, we solve the corresponding penalty equation and show the existence of a solution to the variational inequality. The optimal harvesting and planting policy is characterized in terms of two thresholds for the variational inequality.

This work deals with a prey–predator model in an environment where the carrying capacities are assumed to be variable with time and one species feeds upon the other. Independent harvesting efforts are applied in either species and asymmetrical intraguild predation occurs. A common resource is consumed by two competing species and at the same time predator also consumes the prey. At first we discuss the model under constant carrying capacity and make the conclusion that no limit cycle exists in this case. Then we discuss the model without intraspecific competition. Our main concern is to cover the above mentioned two cases together, i.e. the model with variable carrying capacity and intraspecific competition. We determine the steady states and examine the dynamical behavior. We also analyze the local and global stability of the interior equilibrium by Routh–Hurwitz criterion and a suitable Lyapunov function respectively. A Hopf bifurcation occurs with respect to a parameter which is the ratio of predator’s and prey’s intrinsic growth rate. The possibility of bionomic equilibrium has been considered. The optimal harvest policy is formulated and solved with Pontryagin’s maximum principle. Some numerical simulations are given to explain most of the analytical results.

Gompertz dynamics offer significant applications for the growth of invasive species, cancer modeling, optimal harvesting policies, sustainable yield, and maintaining population levels due to its pattern formation in low‐density cases. This paper examines a widely applicable nonhomogeneous diffusive Gompertz law with zero Neumann boundary conditions, where all coefficients are smooth periodic functions. The analytical approach explains the ubiquitous stability of a time‐periodic solution and seeks the optimal strategy for harvesting under the Gompertz growth law, potentially generalizing the results for many small organisms, including plants and wild populations. The proposed model successfully investigates the dynamics with and without diffusion. Moreover, the spatio‐temporal equation more precisely describes the population’s evolutionary processes using a generalized classical reaction‐diffusion equation. Finally, we observe several potential applications, outlining the optimal strategies for real‐world scenarios and related fields where optimal harvesting is utilized.

Optimal harvesting policy for an age-specific population

<p style='text-indent:20px;'>The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.</p>

We present two models which incorporate prey harvesting to the classical and a modified ratio-dependent predator–prey model with additional food supply to the predators. We analyze the existence and stability of the equilibrium points of both the models and determine the maximum sustainable harvesting effort. The optimal harvesting policy is determined by using Pontryagin’s maximum principle. The results obtained are numerically illustrated. We examine the consequences of providing additional food (as a part of the total harvest effort) to predators in prey harvesting. Our study shows that the system can sustain a much improved optimal prey harvesting rate with additional food supply.

Optimal fishery harvesting rules under uncertainty

Optimal harvesting of a renewable economic resource in a model with Bertalanffy growth law II

The paper presents the study of three species ecological model with Prey N 1 , predator N 2 and competitor to the Predator N 3 and neutral with the predator N 2 with imprecise biological parameters. The model is characterized by a set of first order nonlinear ordinary differential equations. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. Equilibrium points of the model are identified, the local stability is discussed using Routh - Hurwitz criteria and global stability by Liapunov function. The existence of bionomic equilibrium of the system has been discussed and optimal harvesting policy is given using Pontryagin’s maximum principle. The stability analysis is supported by Numerical simulation using Mat lab. Keywords: Prey; Predator; Competitor to the predator; Equilibrium points; interval number, Stability of the equilibrium points; Bionomic Equilibrium; Optimal harvesting policy; Pontryagin’s maximum principle; Numerical simulation using mat lab. DOI : 10.7176/IEL/9-1-04

Harvesting a population sustainably as a resource is a common problem in wildlife and fisheries management. In a typical situation population size and other state variables are monitored at fixed intervals and the resulting estimates are used to determine a quota or harvest effort. Such decisions must be made in the face of a range of uncertainties: environmental variation, an imperfect ability to observe state variables, an imperfect ability to implement management decisions and imperfect knowledge about how the population fluctuates dynamically in response to management actions. In this thesis I explore the effect of these uncertainties on the optimal management of wild populations. In chapters 2 and 3, populations with significant age or stage structure are examined. When individuals of different maturity within a population have significantly different life history traits, the structure of the population and of the harvest taken can have a large effect on the growth rate of the population. In chapter 2 a simple matrix model is used to make observations about the role of demographic structure when the objective is to maintain the population below its carrying capacity. Uncertainty in the structure of the population and the manager's ability to select the harvest structure complicate optimal harvesting. In chapter 3 plausible models are developed for the maintenance of the Atlantic population of Canada geese (Branta canadensis) within acceptable population bounds, given uncertainty about the strength of density dependent population regulation and the limited ability of managers to achieve large harvests. Stochastic dynamic programming is used to determine the optimal harvest strategy under each of the plausible models. It is found that the target long-term population size depends critically on the strength of density dependence. Under the density-independent model, limits to harvest also influence the target long-term population size. Chapter 4 explores the theory of adaptive management. In adaptive management we seek the optimal harvest decision in the presence of model uncertainty. Plausible models are weighted according to the amount of evidence currently supporting them. The optimal harvest decision is obtained by weighting the expected returns under each model. When the population is monitored subsequent to harvesting, the evidence supporting each model can be re-evaluated. In this way the model best describing the system dynamics can be learnt over time (passive adaptive management). Particular harvest decisions may accelerate learning and provide better management in the long term. However these actions are often perceived as risky and so short-term losses must be balanced by long-term benefits (active adaptive management). To test these ideas, a simple population model with an uncertain parameter is constructed. Fixed, passive adaptive and active adaptive harvest strategies are developed using stochastic dynamic programming. It is found that the passive adaptive strategy is `certainty-equivalent', meaning that the current best estimate of the uncertain parameter is used as if it were the true parameter value. Over very long time horizons, the active adaptive strategy probes for information but in the short-term it is actually more precautionary than the certainty-equivalent strategy. The passive and active adaptive strategies perform similarly well in maximising harvest, and both outperform fixed non- adaptive strategies. Two different sets of plausible models produce consistent results, leading to the conclusion that it is important to incorporate model uncertainty, but the specific approach does not critically affect the results. In chapter 5 the problem of optimal adaptive monitoring is considered. The most common approach to harvest management is to use the same monitoring effort at regular intervals to estimate state variables. This approach neglects the large costs often involved in population monitoring, assuming that the level of accuracy achieved is both necessary and sufficient to make the appropriate harvest decision. I take an alternative approach, combining the costs of monitoring and the expected benefits for management in a single framework to determine the level of monitoring accuracy required. Monitoring effort becomes a state-dependent decision at each time interval, determined by prior information about the state variables. This approach is demonstrated using data for a red kangaroo (Macropus rufus) population in South Australia. This document is not a comprehensive treatment of optimal harvesting under uncertainty. However it does indicate the ways in which uncertainty complicates the harvest of wildlife, and its potential effect on optimal harvesting and monitoring decisions.

In this article we present the qualitative properties of solutions of a periodic differential equation that describes dynamics of a renewable resource subjected to strong Allee effect (multiplicative Allee effect) with periodic harvesting in a periodically fluctuating environment. We also present the bio-economic aspects of the considered renewable resource studied from sole owner perspective. An optimal harvesting problem with periodic dynamic constraints has been solved using Pontryagin’s maximum principle. It is interesting to find that the considered optimal control problem admits multiple periodic singular solutions. Techniques are developed to identify the optimal singular solution and the associated optimal singular periodic effort function. Optimal, suboptimal harvest policies and their corresponding approach paths are constructed. It is found that the optimal harvest policies are periodic after some time.

Optimal harvest rates were computed using dynamic programming for mixed-stock fisheries exploiting two stocks of either natural or hatchery origin. Natural stocks were described by a Ricker spawner–recruit relationship and hatchery stocks were described by a rectilinear spawner–recruit relationship. Harvest rates were optimized for both risk-neutral and risk-averse utility functions. For two natural stocks with low productivities, optimal harvest rates generally appeared to favor the stronger stock for a risk-neutral utility function and the weaker stock for a risk-averse utility function. For both utility functions, optimal harvest policy became less sensitive to relative stock strength as the productivity of the stocks increased. When at least one of the stocks was of hatchery origin, optimal harvest policy favored the weaker stock using either utility function.