Optimal harvesting for a predator-prey metapopulation

In this paper we present a deterministic, discrete-time model for a two-patch predator-prey metapopulation. We study optimal harvesting for the metapopulation using dynamic programming. Some rules are established as generalizations of rules for a single-species metapopulation harvesting theory. We also establish rules to harvest relatively more (or less) vulnerable prey subpopulations and more (or less) efficient predator subpopulations.     

Some strategies for harvesting a single species

Then-stage harvesting strategy of Elizarov and Svirezhev is examined. As a result, some important new features appear. A discussion is presented on whether or not one should harvest a species at one time stage or wait until a later time. The paper is concerned with contributions which are primarily mathematical formulations and results for continuous, as well as discrete time, logistic growth of a single species being harvested. Age class structure is ignored.     

Optimal harvesting strategies for timber and non-timber forest products in tropical ecosystems

Harvesting wild plants for non-timber forest products (NTFPs) can be ecologically sustainable–without long-term consequences to the dynamics of targeted and associated species–but it may not be economically satisfying because it fails to provide enough revenues for local people over time. In several cases, the same species can be harvested for NTFP and also logged for timber. Three decades of studies on the sustainability of NTFP harvest for local people’s livelihood have failed to successfully integrate these socio-economic and ecological factors. We apply optimal control theory to investigate optimal strategies for the combinations of non-lethal (e.g., NTFP) and lethal (e.g., timber) harvest that minimize the cost of harvesting while maximizing the benefits (revenue) that accrue to harvesters and the conservation value of harvested ecosystems. Optimal harvesting strategies include starting with non-lethal NTFP harvest and postponing lethal timber harvesting to begin after a few years. We clearly demonstrate that slow growth species have lower optimal harvesting rates, objective functional values and profits than fast growth species. However, contrary to expectation, the effect of species lifespan on optimal harvesting rates was weak suggesting that life history is a better indicator of species resilience to harvest than lifespan. Overall, lethal or nonlethal harvest rates must be <40 % to ensure optimality. This optimal rate is lower than commonly reported sustainable harvest rates for non-timber forest products.     

Optimal patch use and metapopulation dynamics

Summary We compared the metapopulation dynamics of predator—prey systems with (1) adaptive global dispersal, (2) adaptive local dispersal, (3) fixed global dispersal and (4) fixed local dispersal by predators. Adaptive dispersal was modelled using the marginal value theorem, such that predators departed patches when the instantaneous rate of prey capture was less than the long-term rate of prey capture averaged over all patches, scaled to the movement time between patches. Adaptive dispersal tended to stabilize metapopulation dynamics in a similar manner to conventional fixed dispersal models, but the temporal dynamics of adaptive dispersal models were more unpredictable than the smooth oscillations of fixed dispersal models. Moreover, fixed and adaptive dispersal models responded differently to spatial variation in patch productivity and the degree of compartmentalization of the system. For both adaptive dispersal and fixed dispersal models, localized (stepping-stone) dispersal was more strongly stabilizing than global (island) dispersal. Variation among predators in the probability of dispersal in relation to local prey density had a strong stabilizing influence on both within-patch and metapopulation dynamics. These results suggest that adaptive space use strategies by predators could have important implications for the dynamics of spatially heterogeneous trophic systems.     

A new approach for designing disease intervention strategies in metapopulation models

We describe a new approach for investigating the control strategies of compartmental disease transmission models. The method rests on the construction of various alternative next-generation matrices, and makes use of the type reproduction number and the target reproduction number. A general metapopulation SIRS (susceptible–infected–recovered–susceptible) model is given to illustrate the application of the method. Such model is useful to study a wide variety of diseases where the population is distributed over geographically separated regions. Considering various control measures such as vaccination, social distancing, and travel restrictions, the procedure allows us to precisely describe in terms of the model parameters, how control methods should be implemented in the SIRS model to ensure disease elimination. In particular, we characterize cases where changing only the travel rates between the regions is sufficient to prevent an outbreak.     

Optimal harvesting and optimal vaccination

Two optimization problems are considered: Harvesting from a structured population with maximal gain subject to the condition of non-extinction, and vaccinating a population with prescribed reduction of the reproduction number of the disease at minimal costs. It is shown that these problems have a similar structure and can be treated by the same mathematical approach. The optimal solutions have a 'two-window' structure: Optimal harvesting and vaccination strategies or policies are concentrated on one or two preferred age classes. The results are first shown for a linear age structure problem and for an epidemic situation at the uninfected state (minimize costs for a given reduction of the reproduction number) and then extended to populations structured by size, to harvesting at Gurtin-MacCamy equilibria and to vaccination at infected equilibria.     

Optimal harvesting and stability for a two-species competitive system with stage structure

In this paper, we consider a stage-structured competitive population model with two life stages, immature and mature, with a mature population of harvesting. We obtain conditions for the existence of a globally asymptotically stable positive equilibrium and a threshold of harvesting for the mature population. The optimal harvesting of the mature population is also considered.     

Optimal life‐history schedule in a metapopulation with juvenile dispersal

Previous models have predicted that when mortality increases with age, older individuals should invest more of their resources in reproduction and produce less dispersive offspring, as both their future reproductive value and their prospect of competing with their own sib decline. Those models assumed stable population sizes. We here study for the first time the evolution of age‐specific reproductive effort and of age‐specific offspring dispersal rate in a metapopulation with extinction‐recolonization dynamics and juvenile dispersal. Our model explores the evolutionary consequences of disequilibrium in the age structure of individuals in local populations, generated by disturbances. Life‐history decisions are then shaped both by changes with age in individual performances, and by changes in ecological conditions, as young and old individuals do not live on average in the same environments. Lower juvenile dispersal favours the evolution of higher reproductive effort in young adults in a metapopulation with extinction‐recolonization compared with a well‐mixed population. Contrary to previous predictions for stable structured populations, we find that offspring dispersal should generally increase with maternal age. This is because young individuals, who are overrepresented in recently colonized populations, should allocate more to reproduction and less to dispersal as a strategy to exploit abundant recruitment opportunities in such populations.     

Optimal harvesting for a single-species population governed by Gompertz law:Influence of environmental fluctuation and limited harvesting capacity

This work presents an optimal harvesting problem associated with a single-species population governed by Gompertz law in a seasonally fluctuating environment.The influence of environmental fluctuation is accommodated by choosing the coefficients in the differential equation to be periodic functions with the same period and restriction on the harvesting effort is accommodated by considering binding constraints on the control variable.Hence,a linear optimal control problem has been considered where the state dynamics is governed by Gompertz equation and the control variable is subject to the binding constraints.With the help of maximum principle and the concept of blocked intervals,an optimal periodic solution has been obtained which is followed by the construe tion of optimal solution using the theory of most rapid approach.Important results of the study are demonstrated through numerical simulations.     

Optimal harvesting of an age-structured population

Here we investigate the optimal harvesting of an age-structured population. We use the McKendrick model of population dynamics, and optimize a discounted yield on an infinite time horizon. The harvesting function is allowed to depend arbitrarily on age and time and its magnitude is unconstrained. We obtain, in addition to existence, the qualitative result that an optimal harvesting policy consists of harvesting at no more than three distinct ages.     

Optimal harvesting of stochastically fluctuating populations

 We obtain the optimal harvesting plan to maximize the expected discounted number of individuals harvested over an infinite future horizon, under the most common (Verhulst-Pearl) logistic model for a stochastically fluctuating population. We also solve the problem for the standard variants of the model where there are constraints on the admissible harvesting rates. We use stochastic calculus to derive the optimal population threshold at which individuals are harvested as well as the overall value of the population in the sense of the model. We show that except under extreme conditions, the population is never depleted in finite time, but remains in a stationary distribution which we find explicitly. Needless to say, our results prove that any strategy which totally depletes the population is sub-optimal. These results are much more precise than those previously obtained for this problem. Received 24 June 1996; received in revised form 7 April 1997     

Optimal harvesting of fish stocks under a time-varying discount rate

Optimal control theory has been extensively used to determine the optimal harvesting policy for renewable resources such as fish stocks. In such optimisations, it is common to maximise the discounted utility of harvesting over time, employing a constant time discount rate. However, evidence from human and animal behaviour suggests that we have evolved to employ discount rates which fall over time, often referred to as “hyperbolic discounting”. This increases the weight on benefits in the distant future, which may appear to provide greater protection of resources for future generations, but also creates challenges of time-inconsistent plans. This paper examines harvesting plans when the discount rate declines over time. With a declining discount rate, the planner reduces stock levels in the early stages (when the discount rate is high) and intends to compensate by allowing the stock level to recover later (when the discount rate will be lower). Such a plan may be feasible and optimal, provided that the planner remains committed throughout. However, in practice there is a danger that such plans will be re-optimized and adjusted in the future. It is shown that repeatedly restarting the optimization can drive the stock level down to the point where the optimal policy is to harvest the stock to extinction. In short, a key contribution of this paper is to identify the surprising severity of the consequences flowing from incorporating a rather trivial, and widely prevalent, “non-rational” aspect of human behaviour into renewable resource management models. These ideas are related to the collapse of the Peruvian anchovy fishery in the 1970's.     

Optimal muscular coordination strategies for jumping

This paper presents a detailed analysis of an optimal control solution to a maximum height squat jump, based upon how muscles accelerate and contribute power to the body segments during the ground contact phase of jumping. Quantitative comparisons of model and experimental results expose a proximal-to-distal sequence of muscle activation (i.e. from hip to knee to ankle). We found that the contribution of muscles dominates both the angular acceleration and the instantaneous power of the segments. However, the contributions of gravity and segmental motion are insignificant, except the latter become important during the final 10% of the jump. Vasti and gluteus maximus muscles are the major energy producers of the lower extremity. These muscles are the prime movers of the lower extremity because they dominate the angular acceleration of the hip toward extension and the instantaneous power of the trunk. In contrast, the ankle plantarflexors (soleus, gastrocnemius, and the other plantarflexors) dominate the total energy of the thigh, though these muscles also contribute appreciably to trunk power during the final 20% of the jump. Therefore, the contribution of these muscles to overall jumping performance cannot be neglected. We found that the biarticular gastrocnemius increases jump height (i.e. the net vertical displacement of the center of mass of the body from standing) by as much as 25%. However, this increase is not due to any unique biarticular action (e.g. proximal-to-distal power transfer from the knee to the ankle), since jumping performance is similar when gastrocnemius is replaced with a uniarticular ankle plantarflexor.     

Support for a metapopulation structure among mammals

• 1 The metapopulation metaphor is increasingly used to explain the spatial dynamics of animal populations. However, metapopulation structure is difficult to identify in long‐lived species that are widely distributed in stochastic environments, where they can resist extinctions. The literature on mammals may not provide supporting evidence for classic metapopulation dynamics, which call for the availability of discrete habitat patches, asynchrony in local population dynamics, evidence for extinction and colonization processes, and dispersal between local populations.
• 2 Empirical evidence for metapopulation structure among mammals may exist when applying more lenient criteria. To meet these criteria, mammals should live in landscapes as discrete local breeding populations, and their demography should be asynchronous.
• 3 We examined the literature for empirical evidence in support of the classical criteria set by Hanski (1999 ), and for the more lenient subset of criteria proposed by Elmhagen & Angerbjörn (2001 ). We suggest circumstances where metapopulation theory could be important in understanding population processes in mammals of different body sizes.
• 4 The patchy distribution of large (>100 kg) mammals and dispersal often motivate inferences in support of a metapopulation structure. Published studies seldom address the full suite of classical criteria. However, studies on small mammals are more likely to record classic metapopulation criteria than those on large mammals. The slow turnover rate that is typical for medium‐sized and large mammals apparently makes it difficult to identify a metapopulation structure during studies of short duration.
• 5 To identify a metapopulation structure, studies should combine the criteria set by Hanski (1999 ) and Elmhagen & Angerbjörn (2001 ). Mammals frequently live in fragmented landscapes, and processes involved in the maintenance of a metapopulation structure should be considered in conservation planning and management.

     

Stabilization of population dynamics via threshold harvesting strategies

We study two different harvesting/thinning control strategies in the framework of one-dimensional discrete time population models. They have the common feature of considering a threshold population size, commonly called Biomass at the limit, under which the population is not altered, and they differ in how the harvesting/thinning is applied when that threshold is surpassed: one uses the well known proportional feedback control, whereas the other employs the recently proposed target oriented control. We focus on the possibility of applying these strategies to control the chaotic behaviour predicted by some one-dimensional discrete time population models. We discuss the basic properties of both strategies and compare them with other simpler control methods. Particularly, we show that increasing the threshold does not affect, or almost does not affect, a stable exploited population as long as the threshold is lower than the carrying capacity of the system.     

Evaluation of alternate harvesting strategies using experimental microcosms

Experimental evidence to evaluate alternate conservation policies for harvested populations is currently meager. We used populations of the ciliate Tetrahymena thermophila growing in test tube microcosms to experimentally evaluate the effects of alternate harvesting policies in a controlled, replicable setting. Simple density-dependent models were effective in predicting patterns of ciliate population growth in the microcosms. We evaluated several univariate models, finding that a Ricker logistic model was a better predictor of ciliate population dynamics than Gompertz logistic, non-linear logistic, or random walk models. Using the Ricker logistic model as a demographic skeleton, we modeled ciliate population dynamics with respect to three alternate harvesting policies (fixed quota, fixed proportion, and fixed escapement), each conducted at four comparable levels of harvest intensity. The parameterized demographic models predicted that fixed quota harvesting would lead to lower mean ciliate abundance and higher temporal variability in ciliate abundance than fixed proportion or fixed escapement policies, with an appreciable risk of extinction, even under the controlled environmental conditions of our experimental system. For each harvesting policy, the intensity of harvest had demonstrable effects on population density. Population variability was higher for fixed quota harvesting than the other policies. The stochastic demographic model successfully predicted heightened extinction risk in the fixed quota system, relative to the other management treatments. Our experimental evidence lends support to the theoretical prediction that fixed quota harvesting is riskier than fixed proportion or fixed escapement policies.     

Optimal harvesting of a logistic population in an environment with stochastic jumps

Dynamic programming is employed to examine the effects of large, sudden changes in population size on the optimal harvest strategy of an exploited resource population. These changes are either adverse or favorable and are assumed to occur at times of events of a Poisson process. The amplitude of these jumps is assumed to be density independent. In between the jumps the population is assumed to grow logistically. The Bellman equation for the optimal discounted present value is solved numerically and the optimal feedback control computed for the random jump model. The results are compared to the corresponding results for the quasi-deterministic approximation. In addition, the sensitivity of the results to the discount rate, the total jump rate and the quadratic cost factor is investigated. The optimal results are most strongly sensitive to the rate of stochastic jumps and to the quadratic cost factor to a lesser extent when the deterministic bioeconomic parameters are taken from aggregate antarctic pelagic whaling data.Research supported in part by the National Science Foundation under grants MCS 81-01698 and MCS 83-00562.     

Optimal parasite infection strategies: a state-dependent approach

Many macroparasites spend a crucial phase of their life-cycle as free-living infective stages. Despite their importance, however, little theoretical work has considered how evolution may shape the behaviour of these stages. Here, we develop what we believe to be the first stochastic dynamic programming model of parasite life-history strategies to investigate how a trade-off between resource depletion and host encounter rate may shape the optimal infection strategy of a macroparasite. The optimal strategy depends strongly on the probability of host contact and, depending on the relative costs and benefits, macroparasites should adopt either a passive 'ambushing' (sit-and-wait) strategy, an active 'cruising' strategy or a mixed strategy with an initial cruising phase, followed by a switch to ambushing when energy reserves fall to a threshold level. Under no circumstances does the model predict ambush-then-cruise. We use our model to help interpret previously published data on entomopathogenic nematode foraging behaviour, showing how this framework could facilitate our understanding of macroparasite behaviour during this key stage of the life-cycle.