MULTISPECIES AND SINGLE‐SPECIES MODELS OF FISH POPULATION DYNAMICS: COMPARING PARAMETER ESTIMATES

Abstract Population features inferred from single‐species, age‐structured models are compared to those inferred from a multispecies, age‐structured model that includes predator‐prey interactions among three commercially harvested fish species—walleye pollock, Atka mackerel, and Pacific cod—on the Aleutian Shelf, Alaska. The multispecies framework treats the single‐species models and data as a special case of the multispecies model and data. The same data from fisheries and surveys are used to estimate model parameters for both single‐species and multispecies configurations of the model. Additionally, data from stomach samples and predator rations are used to estimate the parameters of the multispecies model. One form of the feeding functional response, predator pre‐emption, was selected using AIC from seven alternative models for how the predation rate changes with the densities of prey and possibly other predators. Differences in estimated population dynamics and productivity between the multispecies and single‐species models were observed. The multispecies model estimated lower mackerel population sizes from 1964–2003 than the single‐species model, while the spawning biomass of pollock was estimated to have declined more than three times faster since 1964 by the multispecies model. The variances around the estimates of spawning biomass were smaller for mackerel and larger for pollock in the multispecies model compared to the single‐species model.     

Food chain model with optimal harvesting in fuzzy environment

A multispecies harvesting model with mutual interactions is formulated based on Lotka–Voltera model with three competing species which are affected not only by harvesting but also by the presence of prey, predator and the third species, which is super predator. In order to understand the dynamics of the system, it is assumed that the super predator follows the logistic growth. Further, there is demand for all the above three species in the market and hence harvesting of all species is performed. We derive the condition for global stability of the system using a suitable Lyapunov function. The possibility of existence of bioeconomic equilibrium is discussed. The optimal harvest policy is studied and the solution is derived under imprecise inflation in fuzzy environment using Pontryagin’s maximal principle. Finally some numerical examples are discussed to illustrate the model.     

ASSESSING UNCERTAINTY IN A MULTISPECIES AGE‐STRUCTURED ASSESSMENT FRAMEWORK: THE EFFECTS OF DATA LIMITATIONS AND MODEL ASSUMPTIONS

Performance of a multispecies age‐structured assessment (MSASA) model in the Gulf of Alaska (GOA) relative to changes in data and model assumptions was examined through simulation exercises. Species included arrowtooth flounder (Atheresthes stomias), Pacific cod (Gadus macrocephalus), walleye pollock (Theragra chalcogramma), Pacific halibut (Hippoglossus stenolepis), and Steller sea lion (Eumetopias jubatus). Age‐specific predation mortality was estimated as a flexible function of predator and prey abundances and fitted to diet data. Simulated data sets were constructed by applying random error to estimates of catch, survey, and diet data from an operating model, whose structure was identical to that of the estimating model. Simulations explored the effects of data variability, mismatched assumptions regarding model structure, and lack of diet data on model performance. Model misspecification and uninformative diet data had the greatest influence on model performance. Given the current emphasis on the development of ecosystem‐based models and management, prioritizing the rigorous sampling of diet data would best facilitate the development of predation models useful to management agencies.     

Ergodicity and threshold behaviors of a predator‐prey model in stochastic chemostat driven by regime switching

This paper deals with a stochastic predator‐prey model in chemostat which is driven by Markov regime switching. For the asymptotic behaviors of this stochastic system, we establish the sufficient conditions for the existence of the stationary distribution. Then, we investigate, respectively, the extinction of the prey and predator populations. We explore the new critical numbers between survival and extinction for species of the dual‐threshold chemostat model. Numerical simulations are accomplished to confirm our analytical conclusions.     

Period doubling cascades of prey–predator model with nonlinear harvesting and control of over exploitation through taxation

The present study investigates a prey predator type model for conservation of ecological resources through taxation with nonlinear harvesting. The model uses the harvesting function as proposed by Agnew (1979) [1] which accounts for the handling time of the catch and also the competition between standard vessels being utilized for harvesting of resources. In this paper we consider a three dimensional dynamic effort prey–predator model with Holling type-II functional response. The conditions for uniform persistence of the model have been derived. The existence and stability of bifurcating periodic solution through Hopf bifurcation have been examined for a particular set of parameter value. Using numerical examples it is shown that the system admits periodic, quasi-periodic and chaotic solutions. It is observed that the system exhibits periodic doubling route to chaos with respect to tax. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. Pontryagin’s Maximum Principle has been used to obtain optimal tax policy to maximize the monetary social benefit as well as conservation of the ecosystem.     

A MICRO‐LEVEL ‘CONSUMER APPROACH’ TO SPECIES POPULATION DYNAMICS

ABSTRACT. In this paper we develop a micro ecosystem model whose basic entities are representative organisms which behave as if maximizing their net offspring under constraints. Net offspring is increasing in prey biomass intake, declining in the loss of own biomass to predators and Allee's law applies. The organism's constraint reflects its perception of how scarce its own biomass and the biomass of its prey is. In the short‐run periods prices (scarcity indicators) coordinate and determine all biomass transactions and net offspring which directly translates into population growth functions. We are able to explicitly determine these growth functions for a simple food web when specific parametric net offspring functions are chosen in the micro‐level ecosystem model. For the case of a single species our model is shown to yield the well‐known Verhulst‐Pearl logistic growth function. With two species in predator‐prey relationship, we derive differential equations whose dynamics are completely characterized and turn out to be similar to the predator‐prey model with Michaelis‐Menten type functional response. With two species competing for a single resource we find that coexistence is a knife‐edge feature confirming Tschirhart's [2002] result in a different but related model.     

Optimal harvesting in a predator-prey model with Allee effect and sigmoid functional response

This work deals with the determination of the optimal harvest policy in an open access fishery in which both prey and predator species are subjected to non-selective harvesting.The model is described by autonomous ordinary differential equation systems, the functional response of the predators is Holling type III and the prey growth is affected by the Allee effect. The catch-rate functions are based on the catch per unit effort (CPUE) or Schaefer’s hypothesis.The problem of determining the optimal harvest policy is solved by using Pontryagin’s maximal principle. The problem here studied is to maximize a cost function representing the present value of a continuous time-stream of revenue of the fishery.     

THE IMPACT OF SCARCITY AND ABUNDANCE IN FOOD CHAINS ON SPECIES POPULATION DYNAMICS

ABSTRACT. The population dynamics in a food chain are derived from a sequence of short‐run equilibria of an ecosystem where predator species demand prey biomass, supply own biomass to their predators and are assumed to behave as if they maximize net biomass intake. Introducing prices as scarcity indicators for the biomass of each species enables us to determine a short‐run ecosystem equilibrium guided by prices. Equilibrium regimes differ with respect to their mix of zero‐priced (= abundant) and positive‐priced (= scarce) species. The population dynamics turn out to vary with the prevailing equilibrium regime. Our analysis yields a richer and more complex population dynamics than the traditional predator‐prey dynamics of the Lotka‐Volterra type.     

THE ENSEMBLE KALMAN FILTER FOR MULTIDIMENSIONAL BIOECONOMIC MODELS

To integrate economic considerations into management decisions in ecosystem frameworks, we need to build models that capture observed system dynamics and incorporate existing knowledge of ecosystems, while at the same time accommodating economic analysis. The main constraint for models to serve in economic analysis is dimensionality. In addition, to apply in long‐term management analysis, models should be stable in terms of adjustments to new observations. We use the ensemble Kalman filter to fit relatively simple models to ecosystem or foodweb data and estimate parameters that are stable over the observed variability in the data. The filter also provides a lower bound on the noise terms that a stochastic analysis requires. In this paper, we apply the filter to model the main interactions in the Barents Sea ecosystem. In a comparison, our method outperforms a regression‐based approach.     

Dynamics of a stochastic three species prey-predator model with intraguild predation

Intraguild predation is ubiquitous in many ecological communities. This paper is concerned with a stochastic three species prey-predator model with intraguild predation. The model involves a prey, an intermediate predator which preys on only prey and an omnivorous top predator which preys on both prey and intermediate predator. First, we show the existence of a unique positive global solution of the model. Then we mainly establish the sufficient conditions for the extinction and persistence in the mean of each population. Moreover, we show that the model is stable in distribution. Finally, some numerical simulations are given to illustrate the main results.     

Optimal control of effort of a stage structured prey–predator fishery model with harvesting

This paper describes a prey–predator fishery model with stage structure for prey. The adult prey and predator populations are harvested in the proposed system. The dynamic behavior of the model system is discussed. It is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account. We have incorporated state feedback controller to stabilize the model system in the case of positive economic interest. Fishing effort used to harvest the adult prey and predator populations is used as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.     

Stochastic persistence and stability analysis of a modified Holling–Tanner model

The article aims to study the basic dynamical features of a modified Holling–Tanner prey–predator model with ratio‐dependent functional response. We have proved the global existence of the solution for the deterministic model. The parametric restriction for persistence of both species is also obtained along with the proof of local asymptotic stability of the interior equilibrium point(s). Conditions for local bifurcations of interior equilibrium points are provided. The global dynamic behavior is examined thoroughly with supportive numerical simulation results. Next, we have formulated the stochastic model by perturbing the intrinsic growth rates of prey and predator populations with white noise terms. The existence uniqueness of solutions for stochastic model is established. Further, we have derived the parametric restrictions required for the persistence of the stochastic model. Finally, we have discussed the stochastic stability results in terms of the first and second order moments. Numerical simulation results are provided to support the analytical findings. Copyright © 2012 John Wiley & Sons, Ltd.     

ECONOMIC IMPACTS OF GLOBAL WARMING: THE CASE OF THE BARENTS SEA FISHERIES

ABSTRACT. . Regional analyses of possible physical and biological effects of global warming in the Barents Sea area have been carried out recently. Based on such studies possible economic impacts of global warming on the Barents Sea fisheries have been quantified, assuming different types of management regimes. The EconSimp2000 model, consisting of the ecosystem model AggMult and the fleet model EconMult have been parameterized based on fleet and catch records from the Norwegian Barents Sea fisheries. The model has been used to study biological and economic impacts of different environmental scenarios representing possible consequences of global warming. The current environmental situation, including normal seasonal and other variations, has been used as a reference scenario. Several biological and economic indicators have been defined in order to evaluate the simulation results of different environmental scenarios and different types of management regimes. The findings support earlier studies where biological and economic impacts of changes in management regime is found to be more pronounced than impacts caused by effects of global warming.     

STABILITY ANALYSIS OF A TRITROPHIC FOOD CHAIN MODEL WITH AN ADAPTIVE PARAMETER FOR THE PREDATOR

Abstract The study of three‐species communities have become the focus of considerable attention, and because the studies of ecological communities start with their food web, we consider a tritrophic food chain model comprised of the prey, the predator, and the super‐predator. The classical assumption of the domino effect is supplemented with an adaptive parameter for the predator (in the absence of prey). Thus, the model exhibits an equilibrium with the predator‐top‐predator steady state, which is a saddle point. Dynamical behaviors such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. The long‐term coexistence of the three interacting species is addressed, and the stability analysis of the model shows that the biologically most relevant equilibrium point is globally asymptotically stable whenever it satisfies a certain criterion. Practical implications are explored and related to real populations.     

Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.     

Stationary distribution and extinction of a stochastic predator–prey model with distributed delay

In this paper, we focus on a stochastic predator–prey model with distributed delay. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the predator population, that is, the prey population is survival and the predator population is extinct.     

A MICROFOUNDATION OF PREDATOR‐PREY DYNAMICS

ABSTRACT. Predator‐prey relationships account for an important part of all interactions betweenspecies. In this paper we provide a microfoundation for such predator‐prey relations in afood chain. Basic entities of our analysis are representative organisms of species modeled similar to economic households. With prices as indicators of scarcity, organisms are assumed to behave as if they maximize their net biomass subject to constraints which express the organisms' risk of being preyed upon during predation. Like consumers, organisms face a ‘budget constraint’ requiring their expenditure on prey biomass not to exceed their revenue from supplying own biomass. Short‐run ecosystem equilibria are defined and derived. The net biomass acquired by the representative organism in the short term determines the positive or negative population growth. Moving short‐run equilibria constitute the dynamics of the predator‐prey relations that are characterized in numerical analysis. The population dynamics derived here turn out to differ significantly from those assumed in the standard Lotka‐Volterra model.     

Asymptotic dynamics of a deterministic and stochastic predator–prey model with disease in the prey species

In this paper, we discuss a predator–prey model with the Beddington–DeAngelis functional response of predators and a disease in the prey species. At first we study permanence and global stability of a positive equilibrium for the deterministic version of the model. Then we include a stochastic perturbation of the white noise type. We analyse the influence of this stochastic perturbation on the systems and prove that the positive equilibrium is also globally asymptotically stable in this case. The key point of our analysis is to choose appropriate Lyapunov functionals. We point out the differences between the deterministic and stochastic versions of the model. Copyright © 2013 John Wiley & Sons, Ltd.     

An eco‐epidemiological predator–prey model where predators distinguish between susceptible and infected prey

A predator–prey model with disease amongst the prey and ratio‐dependent functional response for both infected and susceptible prey is proposed and its features analysed. This work is based on previous mathematical models to analyse the important ecosystem of the Salton Sea in Southern California and New Mexico where birds (particularly pelicans) prey on fish (particularly tilapia). The dynamics of the system around each of the ecologically meaningful equilibria are presented. Natural disease control is considered before studying the impact of the disease in the absence of predators and the interaction of predators and healthy prey and the disease effects on predators in the absence of healthy prey. Our theoretical results are confirmed by numerical simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

A MODEL FOR THE BIOECONOMIC EVALUATION OF MARINE PROTECTED AREA SIZE AND PLACEMENT IN THE NORTH SEA

ABSTRACT. The use of marine protected areas (MPAs) as a basic management tool to limit exploitation rates in marine fisheries has been widely suggested. Models are important in predicting the consequences of management decisions and the design of monitoring programs in terms of policy goals. However, few tools are available that consider both multiple fleets and ecosystem scale dynamics. We use a new applied game theory tool, Ecoseed, that operates within a temporally and spatially explicit biomass dynamics model, Ecopath with Ecosim, to evaluate the efficacy of marine protected areas in the North Sea in both ecological and economic terms. The Ecoseed model builds MPAs based on the change in values of predicted economic rents of fisheries and the existence value of biomass pools in the ecosystem. We consider the market values of four fisheries operating in the North Sea: a trawl fishery, a gill net fishery, a seine fishery, and an industrial (reduction) fishery. We apply existence values, scaled such that their aggregate is similar to the total fishery value, to six biomass pools of concern: juvenile cod, haddock, whiting, saithe, seals, and the collective pool ‘Other predators’ that include marine mammals. Four policy options were considered: to maximize the rent only; to maximize the existence values only; to maximize the sum of the rent and existence values; and, finally, to maximize the sum of the rent and the existence values, but excluding only the trawl fleet from the MPA. The Ecoseed model suggests that policy goals that do not include ecological considerations can negatively impact the rents obtained by the different fishing sectors. The existence values will also be negatively impacted unless the MPA is very large. The Ecoseed model also suggests that policy goals based solely on existence values will negatively impact most fisheries. Under policy options that included ecological considerations, maximum benefits were derived from an MPA that covered 25–40% of the North Sea, placed along the southern and eastern coasts. Finally, the Ecoseed model suggests that an exclusion of the trawl fishery only from the MPA can provide small‐to‐substantial positive impacts to most species and fleets; this relative impact depends on level of interaction between the trawl fleet and the other fleets target species (e.g., through bycatch).