Journal of Marine Biology

Volume 2015, Article ID 580520, 8 pages

http://dx.doi.org/10.1155/2015/580520

## On the Interplay between Cannibalism and Harvest in Stage-Structured Population Models

Harstad University College, Havnegata 5, 9480 Harstad, Norway

Received 23 March 2015; Accepted 3 June 2015

Academic Editor: Garth L. Fletcher

Copyright © 2015 Arild Wikan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By use of a nonlinear stage-structured population model the role of cannibalism and the combined role of cannibalism and harvest have been explored. Regarding the model, we prove that in most parts of parameter space it is permanent. We also show that the transfer from stability to nonstationary dynamics always occurs when the unique stable equilibrium undergoes a supercritical Neimark-Sacker (Hopf) bifurcation. Moreover, the dynamic consequences of catch depend not only on which part of the population (immature or mature) is exposed to increased harvest pressure but also on which part of the immature population (newborns, older immature individuals) suffers from cannibalism. Indeed, if only newborns are exposed to cannibalism an enlargement of harvest pressure on the mature part of the population may act in a stabilizing fashion. On the other hand, whenever the whole immature population is exposed to cannibalism there are parts in parameter space where increased harvest on the mature population acts in a destabilizing fashion.

#### 1. Introduction

Several species of commercial interest have been overexploited throughout the years. Among them we find salmon species like the capelin in the Barents Sea as well as several cod stocks around the world. Lots of other examples may be obtained in [1, 2]. The global production of marine capture fisheries from around 19 million tonnes in catch in the 1950s has increased to around 80 million tonnes annually since the mid-1980s; see [3], and as documented in [4], there are species that have become extinct or almost extinct.

Another characteristic feature of lots of populations is that they oscillate. There may be a substantial difference in biomass from one year to another. Regarding fish populations there may be several causes for such fluctuations. One important factor is environmental changes, for example, changes in current systems which may have a crucial impact of newborns particularly; see [5] and references therein. The presence of one or several predator populations plays an important role too; see [6–8] or [9]. Internal factors like recruitment and cannibalism often act differently with respect to stability properties. While increased recruitment may give birth to chaotic oscillations, there is a tendency that increased cannibalism seems to stabilize the dynamics but not always (cf. the discussion in [10–14]). Finally, change in fishing patterns may also influence the dynamics as accounted for in [9, 15, 16].

The purpose of this paper is to study the combined effects of recruitment, cannibalism, and harvest, and in doing so we apply a discrete stage-structured population model. In the next section we present the model and discuss its properties. An analysis of the impact of increasing recruitment and cannibalism is presented in Section 3. In Section 4 different harvest strategies are included as well in the model, and finally in Section 5 we summarize and discuss results.

#### 2. The Model

At time we split the population into two separate parts, an immature part and a mature part , and we further assume that the relation between the subpopulations at two consecutive time steps may be expressed by a system or difference equations:which we also may write on matrix formwhere andThe meaning of the entries in (1) and (2) is as follows: , , is the fecundity, that is, the number of newborns per adult. , , is the fraction of the immature population that becomes mature one time unit later. , , is the part of the mature population which still lives one year later. The nonlinearities are of Ricker type and the parameters , , , will be referred to as cannibalism parameters. Consequently, the fecundity is reduced by the factor due to cannibalism from the mature population, and in the same way, the remaining part of the immature population is reduced by a factor . We assume no cannibalism pressure on . and , , , are the fractions of each subpopulation which is removed through fishery, respectively.

Obviously, models (1) and (2) have a trivial equilibriumDefine the inherent net reproductive number asThen by stability analysis, it is straightforward to show that is stable provided . Therefore, we will in the rest of the paper assume that . Following [17], a population model is said to be permanent if there exist and such thatwhere is the total population. Hence, if a population model is permanent the total population density neither explodes nor goes to zero. Regarding our models (1) and (2) we have the following result.

Theorem 1. *Assume . Then models (1) and (2) are permanent.*

*Proof. *According to (6) we must show that the total population neither goes to zero (i) nor explodes (ii). Regarding (i), we have already shown that guarantees that the origin is a repellor. Moreover, the restriction on the parameters and functions given in (2) ensures that is irreducible and that is nonnegative for all . Consequently, (1) and (2) are forward invariant. It remains to prove (ii) and in order to do that we need to show that there exists a compact set such that for all there exists satisfying for all . To this end, assume , Thenand by inductionThen there exists such that for Further, in case of , (1) and (2) also giveand once again (by induction) we find that for Finally, take and . Then, for , , , and we are done.

Models (1) and (2) have also a nontrivial equilibrium . Indeed, from (1)and is the solution of the equation , whereClearly, and in case of sufficiently large we conclude that . Moreover,Hence, there exists such that . Consequently, the nontrivial equilibrium is unique.

Let be the Jacobian of (1) and (2) evaluated at . Then is stable whenever the following inequalities hold: After some time-consuming calculations it is possible to rewrite (15a), (15b), and (15c) asIf both eigenvalues of the linearization of (1) and (2) are located inside the unit circle in the complex plane, (16a), (16b), and (16c) hold and is stable. The left hand side of (16a) fails to be positive when an eigenvalue crosses the unit circle through 1 and a saddle-node bifurcation occurs. (16b) fails when . This gives birth to a flip (period doubling) bifurcation; hence when fails to be stable the result is a 2-period orbit. (16c) fails when becomes a complex number located on the boundary of the unit circle. In this case the equilibrium will undergo a Neimark-Sacker (Hopf) bifurcation at instability threshold, and as we penetrate into the unstable parameter region, quasiperiodic orbits restricted to an invariant curve will be the outcome provided the bifurcation is of supercritical type.

#### 3. Recruitment and Cannibalism

First, let us focus on the dynamics in case of no harvest (i.e., ).

Assume , which means that only newborns are exposed to cannibalism.

Then and the nontrivial equilibrium becomesMoreover, criteria (16a) and (16b) degenerate to , , respectively, and both of them are obviously valid. Inequality (16c) may be expressed asHence, fails to be stable when (18) becomes an equality and a Neimark-Sacker bifurcation occurs.

As is well known bifurcations may be of both supercritical and subcritical nature. If a fixed point will undergo a supercritical bifurcation it means that an eigenvalue must cross the unit circle outwards at instability and in the Neimark-Sacker case that an attracting (stable) quasiperiodic orbit restricted on an invariant curve is created beyond the threshold.

Now, considering (1) and (2) we increase through an increase of . Thus the bifurcation takes place at . Moreover, the eigenvalues may be expressed asand an easy computation shows that(evaluated at ) which proves that leaves the unit circle at threshold.

In order to show that the quasiperiodic orbit is stable when , small, we first write (1) and (2) on complex form and then through a series of near identity transformations (normal form calculations) formally express it as(for details, cf. [18] or [19]) and the sign of will determine the nature of bifurcation, implies supercritical, and implies subcritical.

Regarding (1) and (2) we may partly rest upon findings obtained in [13] and express aswhich is negative whenever , . Consequently, the bifurcation is supercritical.

In order to visualize the findings above we show in Figure 1(a) an orbit which converges towards the stable equilibrium , . Figure 1(b) shows the situation after the supercritical bifurcation . The dynamics is restricted to an attracting invariant curve and on that curve (1) and (2) act as a circle map which rotates points around the curve with an irrational winding number. This scenario persists in a large interval but eventually chaos is introduced when the curve starts to break up. This is displayed in Figure 1(c).