Subscribe Now Subscribe Today
Research Article
 

Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery



T.K. Kar and H. Matsuda
 
Facebook Twitter Digg Reddit Linkedin StumbleUpon E-mail
ABSTRACT

In this research we study the dynamics of a prey-predator system, where predator has two stages, a juvenile stage and a mature stage and are harvested by two different groups of fishermen. The existence of possible steady states along with their local and global stability is discussed. We obtained the condition for permanence of the system. We analyzed optimum management of these fisheries and determined optimal levels of stock, effort and catch using a hypothetical set of parameter values.

Services
Related Articles in ASCI
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

T.K. Kar and H. Matsuda, 2007. Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery. Research Journal of Environmental Sciences, 1: 35-46.

DOI: 10.3923/rjes.2007.35.46

URL: https://scialert.net/abstract/?doi=rjes.2007.35.46

INTRODUCTION

Many consumer species go through two or more life stages as they proceed from birth to death. However, the majority of the models in the literature always assumed that during the whole life histories, each individual admits the same density-dependent rate as well as the identical ability to bear and compete with other species, which clearly unrealistic. In many species, only the mature predator food on the prey and its immature are too weak to food on the prey. Therefore, it is practical to introduce the stage-structure into the competitive or prey-predator models. Some of the stage structure models can be found in Aiello and Freedman (1990), Aiello et al. (1992), Freedman and Wu (1991), Gambell (1985), Landahl and Hanson (1975), Wood et al. (1989) and the references therein. A good overview on stage-structured models can be found in the recent book by Murdoch et al. (2003). Recently, papers like Bosch and Gabriel (1997), Kar (2003), Kar and Matsuda (2006), Zhang et al. (2000) and Wikan (2004) study the stage structure of species with or without time delays.

Harvesting has also a strong impact on a dynamic property of a population. Depending on the nature of applied harvesting strategy, the long-run stationary density of the population may be significantly smaller than the long run stationary density of a population in the absence of harvesting. In the absence of harvesting, a population can be free from extinction risk; however, harvesting can lead to the incorporation of a positive extinction probability and therefore, to potential extinction in a finite time. If a population is subject to a positive extinction rate then harvesting can drive the population density to a dangerously low level at which extinction becomes sure no matter how the harvester affects the population afterwards.

A fundamental issue in population biology is what are the minimal conditions to ensure the long term survivorship for all of the interacting components. When these conditions are meet the interacting populations are said to persist or coexist. Permanence corresponds to the existence of a positive attractor that attracts all positive population trajectories. Permanence implies positive population trajectories can recover from large perturbations of the state variables. This question is of particular interest to fishing managers. If it is known that the system exhibits such a permanent behaviour, then ecological planning based on a fixed eventual population can be carried out. Realizing the problem we have obtained the conditions for permanence of the solutions of our system considered.

Though the application of bioeconomics has usually limited to the determination of Total Allowable Catch (TAC), the sharing of the TAC between heterogeneous fisher groups is an important issue to be considered. The fact that different fisher groups harvest different cohorts within fish stocks and thereby have different effects upon both stocks growth and the economics of the fishery, is not taken into account. Political determination of harvest shares has bioeconomic effects, for instance, in the shape of reduced payoffs from the fishery or even extinction. This study is focused on the optimal steady state fish stock level and hence on the steady state Total Allowable Catch (TAC). This is done using a cooperative game theoretic approach to the issue of sharing the harvest. We assume that the harvest strategies of the two vessel groups are determined by their existing technologies and their respective fishing grounds. Thus, the overall optimal sharing of the resource can be determined, after deciding the weights that should be given to the two parties preferences.

THE MODEL

The ecosystem in our model consists of a prey-predator system. We assume that the prey population (denoted by N1) is subjected to a logistic growth condition. The predator population is divided into two stage groups: juvenile predators (denoted by N2) and adult predators (denoted by N3). Here we also assume that only adult predators are capable of preying on the prey species and that the juvenile predators live on their parents. For example, the Chinese Alligator can be regarded as a stage-structured species since the mature is more than 10 years old and can also be regarded as a predator because almost all aquatic animals are the chief food of the Chinese Alligator. Another key and somewhat novel feature of our model is to account for the universally prevalent intra-specific competition in the consumer growth dynamic (Kuang et al., 2003). This intra-specific competition is assumed to induce additional instantaneous deaths only to the adult population and the increased death rate is proportional to the square of the adult population. Holling type II functional response probably a better description of the actual predation seen in nature but for simplicity, we assume the simplest description.

With this assumptions, we have the following plausible two stage prey-predator interaction model:

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(1)

Here r1 is the specific growth rate of the prey and k is its carrying capacity. α is the predation parameter; m is the conversion factor; r3 is the death rate of mature predator species; γ is the proportionality constant of transformation of immature to mature predators; r2 = μ + γ, where μ is the death rate; β is the birth rate of the immature populations.

We can reduce the number of parameters by making the following transformation of variables:

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Making these changes, system (1) assume the much simpler form

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(2)

where

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Now we assume that two groups of fishermen targeting two sub-stocks consist of different stage groups of predators, with mature fish in one sub-stock and immature fish in the other. Under this assumption system (2) becomes

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(3)

Here q1E1x2 and q2E2x3 are based on the catch-per-unit-effort hypothesis (Clark, 1990), where q1and q2 are catchability co-efficients, E1 and E2 are harvesting efforts.

We are not making any case study but the North-East Atlantic cod fishery is a good example for it. The technology that the trawlers and coastal vessels utilize is different, as are the areas of fishing activity and therefore size of harvested fish. The trawlers catch fish of a smaller size than the coastal vessels, as the older fish tend to migrate in to the coast to spawn.

The Iberoatlantic Hake fishery is another example of multifleet fisheries. In this fishery, the species is caught using several fishing methods, particularly trawling, longlining and fixed gillnetting. Specially, trawling acts intensely on younger individuals, whereas the other fishing methods mainly affect more mature specimens.

EQUILIBRIA AND STABILITY ANALYSIS

System (3) has to be analyzed with the following initial conditions : x1(0) > 0, x2(0) > 0 and x3(0) > 0. We observe that the right-hand side of the system (3) is smooth function of the variables (x1, x2, x3) and the parameters, as long as these quantities are non-negative, so local existence and uniqueness properties hold in the positive octant. The state space for system (3) is in the positive octant, {(x1, x2, x3): x1 > 0, x2 > 0 and x3 > 0}, which is clearly an invariant set, since the vector field on the boundary does not point to the exterior. Our next result concerns the existence of equilibrium points.

We find the steady states of system (3) by equating the derivatives on the left hand sides to zero and solving the resulting algebraic equations. This gives three possible steady states, namely, P0(0, 0, 0), P1(a, 0, 0) and P2(x1*, x2*, x3*) where

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Here we want to remark that there exists another equilibrium in the absence of prey if e>c. But it is not realistic since prey is the only source of food for the predator. So throughout the paper we assume that
Therefore, P2 is feasible if

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
hold.

Particularly we are interested on the interior equilibrium point P2(x*1, x*2, x*3) for its usual importance.

Next we consider first the local stability of the equilibria. The variational matrix of the system (3) is given by

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Now,

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

The diagonalization of the Jacobian matrix M (0, 0, 0), yields the following characteristic equation:

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Which shows that P0(0, 0, 0) is unstable.

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Characteristic equation of M(a, 0, 0) is

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

∴P1(a, 0, 0) is locally asymptotically stable for c>e + ad.

Now.

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

The characteristic equation of M(x*1, x*2, x*3) is λ3 + Aλ2 + Bλ + C = 0, where

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Obviously, AB-C>0. According to Routh-Hurwitz criteria, P2(x*1, x*2, x*3) is locally asymptotically stable if

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
hold.

Now we shall discuss the condition of global stability, permanence and extinction of system (3). At first, we give the following notations and definitions

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Definition 1
An equilibrium point P2(x1, x2, x3) is said to be globally asymptotically stable in Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery if it is locally asymptotically stable and all trajectories in converges to P(x1, x2, x3).

Lemma 1

If, Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery then the equilibrium P1(a, 0, 0) is globally asymptotically stable in Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery.

If, Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery then the only interior equilibrium point P2(x*1, x*2, x*3) is globally asymptotically stable in Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery.

Proof

We construct the following Lyapunov function


Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

where αi, i = 1, 2, 3 are positive constants to be determined in the subsequent steps.

Calculating the derivative of V1 along each solution of (3), we have

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Let and α3 = 1.

Then,

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

in Int. Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery, for c≥e+ad.

Therefore, by Lyapunov-LaSalle (Hale, 1969), it follows that P1 is locally asymptotically stable and all trajectories starting in Int. Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery approaches to P1 as t goes to infinity. This establishes the global asymptotic stability (Fig. 1).

Let us take the Lyapunov function


Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

where αi, i = 1, 2, 3 are positive constants to be determined in the subsequent steps.

Calculating the derivative along each solution of (3), we have

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
Fig. 1:

Phase space trajectories of system (3) beginning with different initial states. It is seen that Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery is a global attractor, where a = 3, b = 4, c = 2.5, d = 3.0, e = 0.2, f = 0.7, q1 = 0.3, q2 = 0.5, E1 = 1.5, E2 = 2


Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Let and .

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

By similar arguments as we have used for P1, we may state that P2(x*1, x*2, x*3) is globally asymptotically stable if hold.

Definition 2
System (3) is said to be permanent if there are positive constants m and M such that each positive solution x(t, x0) of (3) with initial condition Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery satisfies

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Definition 3
The ith species of system (3) is said to be extinctive if each positive solution x(t, x0) of (3) with initial condition Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery satisfies

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Combining all these results we get the following theorem.

Theorem 1

The predator species of system (3) is extinctive and the prey species is not extinctive if and only if c+q2E2≥e+ad hold.

Both prey and predator species are permanent if and only if


Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

.Proof
By Definitions 1 and 2 and the Lemma 1, we can easily prove the Theorem.

THE BIOECONOMIC ENVIRONMENT

Once the process of harvesting the resource is started, the problem of management of the fisheries can be viewed in terms of rent maximization, as is the case of several theories of fisheries economics. In this section, the goal is to find the efforts E1 and E2 to maximize a weighted average of the objective functionals obtain from immature and mature predator fish, respectively.

Let II1 = (p1q1x2- c1)E1 and II2 = (p2q2x3- c2)E2

represent the net revenues for the immature and mature species, respectively.

Thereby the present values of the two sub-stocks are, respectively

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

where δ is the discount rate, pi is the unit price of the resource, ci is the unit cost of harvesting the resource substock 1 and 2.

The aim of the social manager would be to select the efforts Ei and stock level to maximize a weighted average of their objective functional, PV. The weights β and (1-β) indicate how much weight is given to the objective functionals. For a given β∈0[0, 1], the management objective functional translates into maximize

PV = βPV1+(1-β)PV2

subject to the stock dynamics given by Eq. 3 and to the control constraints 0 &le Ei &le Eimax. Given the structure of our problem, we arrive at two equations that implicitly define the optimal (steady state) equilibrium for the doubly singular solution (Appendix):

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(4)
Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(5)

The solutions to Eq. 4 and 5 are pursued numerically. From these solutions, we can determine the optimal equilibrium harvest of the two sub-stocks and efforts.

Remark

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

and

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
Fig. 2:

Phase space trajectories of system (3) corresponding to the optimal harvesting efforts. It is seen that the optimal equilibrium point (0.18, 4.59, 1.62) is also a global attractor

This implies that, for each species, the user cost of harvest per unit effort must equal the discounted value of the future marginal profit of effort at the steady state effort level.

For simulation let us take a = 3, b = 4, c = 2.5, d = 3.0, e = 0.2, f = 0.7, q1 = 0.3, q2 = 0.5, c1 = 7, c2 = 8, p1 = 90, p2 = 100, δ = 0.05 in appropriate units.

For the above values of the parameters, optimum harvest is attained for β=0.4. This implies that the highest discounted total profit is achieved when about 40% of the harvest is taken from juvenile fish and 60% taken from the mature fish. The optimum harvesting efforts are and Optimum equilibrium is (0.18, 4.59, 1.62) and discounted profit is 605.7 (Fig. 2).

CONCLUDING REMARKS

An important and one of the interesting questions in mathematical ecology is persistence or permanence, which ensures the survival of biological species and exclude extinction of species for all positive initial conditions. The question of permanence of biological species is of particular interest to fishery. If it known that a system exhibits such a permanent behaviour, then ecological planning based on a fixed eventual population can be carried out. Realizing the problem we have obtained the conditions for persistence of the solutions of our systems.

We have studied how the weights should be divided between two fleets to maximize the harvesting returns from the fishery. We are aware of the simplification involved in converting a highly complex situation into a theoretical model. However, the model that has been developed in the current paper would be a guidelines for the future research.

Before ending this article, we would like to mention that there is still tremendous amount of work to do in this model. For example,

One can consider the stage-structure of prey population.

Gestation period for predator is also an important characteristic to be considered. We leave it for future considerations.

Uncertainties is another important point to be considered.

APPENDIX

The Hamiltonian for the weighted average objective function is

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

where and are adjoint variables.

The control variables E1 and E2 appear linearly in the Hamiltonian function H. Therefore, optimal control will be a combination of bang-bang control and singular control. The optimal control Ei(t) which maximizes H, must satisfy the following conditions:

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

For the singular control we have,

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Now

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(A.1)

and

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(A.2)

We intend to derive here an optimal equilibrium solution of the problem. Since we are considering an equilibrium solution, x, y and E are to be treated as constants in the subsequent steps.

Now the adjoint equations are

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(A.3)

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(A.4)

and

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery
(A.5)

Substituting in (A.4) we get

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Again substituting and in (A.5) we get

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

Substituting in (A.3) we get

Image for - Permanence and Optimization of Harvesting Return: A Stage-Structured Prey-Predator Fishery

ACKNOWLEDGMENT

Authors would like to thank Japan Society for the Promotion in Science (JSPS) for financial support of this research (P05109).

REFERENCES

1:  Aiello, W.G. and H.I. Freedman, 1990. A time delay model of single species growth with stage structure. Math. Biosci., 101: 139-153.
CrossRef  |  PubMed  |  Direct Link  |  

2:  Aiello, W.G., H.I. Freedman and J. Wu, 1992. Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Applied Math, 52: 855-869.
Direct Link  |  

3:  Van den Bosch, F. and W. Gabriel, 1997. Cannibalism in an age-structured predator-prey system. Bull. Math. Biol., 59: 551-567.
CrossRef  |  Direct Link  |  

4:  Clark, C.W., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources. 2nd Edn., Wiley, New York

5:  Freedman, H.I. and J. Wu, 1991. Persistence and global asymptotical stability of single species dispersal models with stage structure. Quart. Applied Math, 49: 351-371.
Direct Link  |  

6:  Gambell, R., 1985. Birds and Mammals-Antarctic Whales, in Antarctica. Pergamon, New York, pp: 223-241

7:  Hale, J.K., 1969. Ordinary Differential Equations. Wiley Interscience, New York

8:  Kar, T.K., 2003. Selective harvesting in a prey-predator fishery with time delay. Math. Comp. Model., 38: 449-458.
Direct Link  |  

9:  Kar, T.K. and H. Matsuda, 2006. Controllability of a harvested prey-predator system with time delay. J. Biol. Syst., 14: 1-12.

10:  Kuang, Y., W. Fagan and I. Loladze, 2003. Biodiversity, habitat area, resource growth rate and interference competition. Bull. Math. Biol., 65: 497-518.
CrossRef  |  Direct Link  |  

11:  Landahl, H.D. and B.D. Hanson, 1975. A three stage population model with cannibalism. Bull. Math. Biol., 37: 11-17.

12:  Murdoch, W.W., C.J. Briggs and R.M. Nisbet, 2003. Consumer-Resource Dynamics. Princeton University Press, Princeton

13:  Wikan, A., 2004. Dynamical consequences of harvest in discrete age-structured population models. J. Math Biol., 49: 35-55.
Direct Link  |  

14:  Wood, S.N., S.P. Blythe, W.S.C. Gurney and R.M. Nisbet, 1989. Instability in mortality estimation schemes related to stage-structure population models. IMA J. Math. Applied Med. Biol., 6: 47-68.

15:  Zhang, X., L. Chen and A.U. Neumann, 2000. The stage-structure predator-prey model and optimal harvesting policy. Math. Biosci., 168: 201-210.
Direct Link  |  

©  2021 Science Alert. All Rights Reserved