OPTIMAL IMPULSIVE HARVEST POLICY FOR AN AUTONOMOUS SYSTEM

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

Optimal impulsive control in periodic ecosystem

<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>

In this paper, the optimization problem of harvesting for a stochastic predator-prey model with S-type distributed time delays (which contain both discrete time delays and continuously distributed time delays) is studied by using ergodic method. Sufficient and necessary conditions for the existence of optimal harvesting strategy are obtained. Moreover, the optimal harvesting effort (OHE, for short) and the maximum of expectation of sustainable yield (MESY, for short) are given. Some numerical simulations are introduced to illustrate our main results.

Optimal harvesting policy for stochastic Logistic population model

Impulsive harvesting and by-catch mortality for the theta logistic model

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

Harvesting is one of the ways for humans to realize economic interests, while unrestricted harvesting will lead to the extinction of populations. This paper proposes a predator–prey model with impulsive diffusion and transient/nontransient impulsive harvesting. In this model, we consider both impulsive harvesting and impulsive diffusion; additionally, predator and prey are harvested simultaneously. First, we obtain the subsystems of the system in prey extinction and predator extinction. We obtain the fixed points of the subsystems by the stroboscopic map theories of impulsive differential equations and analyze their stabilities. Further, we establish the globally asymptotically stable conditions for the prey/predator-extinction periodic solution and the trivial solution of the system, and then the sufficient conditions for the permanence of the system are given. We also perform several numerical simulations to substantiate our results. It is shown that the transient and nontransient impulsive harvesting have strong impacts on the persistence of the predator–prey model.

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.

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 paper, we consider a stochastic Gilpin-Ayala model under regime switching. Obtain the optimal harvesting effort and the maximum sustained yield by investigating the condition of average boundness of the system, and the ergodicity of the Markov chain. Also, through an example, we have proved our conclusion.

Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations

Asymptotic behavior of the principal eigenvalue for cooperative periodic-parabolic systems and applications

Existence and stability of time periodic traveling waves for a periodic bistable Lotka–Volterra competition system

In this paper, discrete-time analogues of continuous-time neural networks with continuously distributed delays and periodic inputs are investigated without assuming Lipschitz conditions on the activation functions. The discrete-time analogues are considered to be numerical discretizations of the continuous-time networks and we study their dynamical characteristics. By employing Halanay-type inequality, we obtain easily verifiable sufficient conditions ensuring that every solutions of the discrete-time analogue converge exponentially to the unique periodic solutions. It is shown that the discrete-time analogues inherit the periodicity of the continuous-time networks. The results obtained can be regarded as a generalization to the discontinuous case of previous results established for delayed neural networks possessing smooth neuron activation.