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 t we split the population xt into two separate parts, an immature part x1,t and a mature part x2,t, and we further assume that the relation between the subpopulations at two consecutive time steps may be expressed by a system or difference equations:(1)x1,t+1=Fe-β1x2,tx2,t+1-μ1e-β2x2,tx1,t1-h1,x2,t+1=μ1x1,t+1-μ2x2,t1-h2which we also may write on matrix form(2)x_t+1=Axx_t,where x_=(x1,x2)T and(3)Ax=1-μ11-h1e-β2x2Fe-β1x21-h1μ11-h21-μ21-h2.The meaning of the entries in (1) and (2) is as follows: F, F>0, is the fecundity, that is, the number of newborns per adult. μ1, 0<μ1<1, is the fraction of the immature population that becomes mature one time unit later. 1-μ2x2,t, 0<μ2<1, is the part of the mature population which still lives one year later. The nonlinearities are of Ricker type and the parameters βi, βi≥0, i=1,2, will be referred to as cannibalism parameters. Consequently, the fecundity is reduced by the factor e-β1x2 due to cannibalism from the mature population, and in the same way, the remaining part of the immature population 1-μ1x1 is reduced by a factor e-β2x2. We assume no cannibalism pressure on x2. h1 and h2, 0≤h1, h2≤1, are the fractions of each subpopulation which is removed through fishery, respectively.
Obviously, models (1) and (2) have a trivial equilibrium(4)x~1,x~2=0,0.Define the inherent net reproductive number R0 as(5)R0=1-h11-h2μ1F1-1-μ11-h11-1-μ21-h2.Then by stability analysis, it is straightforward to show that x~1,x~2 is stable provided R0<1. Therefore, we will in the rest of the paper assume that R0≥1. Following [17], a population model is said to be permanent if there exist δ>0 and D>0 such that(6)δ<limt→∞infxt≤limt→∞supxt<D,where xt=x1,t+x2,t 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 R0>1. 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 R0>1 guarantees that the origin is a repellor. Moreover, the restriction on the parameters and functions given in (2) ensures that A0 is irreducible and that Ax is nonnegative for all (x1,x2)∈R+2. Consequently, (1) and (2) are R+2∖0 forward invariant. It remains to prove (ii) and in order to do that we need to show that there exists a compact set X⊂R+2 such that for all x_0∈R+2 there exists tM=tMx_0 satisfying xt∈X for all t≥tM. To this end, assume exp-β1x2,tx2,t≤K0, Then(7)x1,t+1≤1-μ1x1,t+FK0and by induction(8)x1,t≤1-μ1tx1,0+FK0μ1.Then there exists tA=tAx1,0 such that for t>tA(9)x1,t+1≤2FK0μ1=K1.Further, in case of t>tA, (1) and (2) also give(10)x2,t+1≤μ1K1+1-μ2x2,tand once again (by induction) we find that for t>tB(x2,0)(11)x2,t≤2μ1K1μ2=K2.Finally, take tM=maxtA,tB and K=maxK1,K2. Then, for t≥tM, x1,t≤K, x2,t≤K, and we are done.
Models (1) and (2) have also a nontrivial equilibrium (x1∗,x2∗). Indeed, from (1)(12)x1∗=1-1-μ21-h2μ11-h2x2∗and x2∗ is the solution of the equation g(x2)=0, where(13)gx2=μ1F1-h11-h21-1-μ21-h2e-β1x2+1-h11-μ1e-β2x2-1.Clearly, g′(x2)<0 and in case of x2 sufficiently large we conclude that g(x2)<0. Moreover,(14)limx2→0gx2=R0-11-1-μ11-h1>0.Hence, there exists x2=x2∗ such that g(x2∗)=0. Consequently, the nontrivial equilibrium (x1∗,x2∗) is unique.
Let J be the Jacobian of (1) and (2) evaluated at (x1•,x2•). Then (x1∗,x2∗) is stable whenever the following inequalities hold: (15a)1-trJ+J>0,(15b)1+trJ+J>0,(15c)1-J>0.After some time-consuming calculations it is possible to rewrite (15a), (15b), and (15c) as(16a)1-1-μ21-h2·β1-1-μ11-h1β1-β2e-β2x2∗x2∗>0,(16b)21-μ21-h2+1-1-μ21-h2β1x2∗+1-μ11-h1·2-1-1-μ21-h2β1-β2x2∗e-β2x2∗>0,(16c)2-1-μ21-h2-1-1-μ21-h2β1x2∗-e-β2x2∗1-h11-μ1·1-1-1-μ21-h2β1-β2x2∗>0.If both eigenvalues λ of the linearization of (1) and (2) are located inside the unit circle in the complex plane, (16a), (16b), and (16c) hold and (x1∗,x2∗) 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 λ=-1. This gives birth to a flip (period doubling) bifurcation; hence when (x1∗,x2∗) fails to be stable the result is a 2-period orbit. (16c) fails when λ 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., h1=h2=0).
(1) Assume β1≠0, β2=0 which means that only newborns are exposed to cannibalism.
Then R0=F/μ2 and the nontrivial equilibrium becomes(17)x1∗,x2∗=μ2μ11β1lnR0,1β1lnR0.Moreover, criteria (16a) and (16b) degenerate to μ1μ2lnR0>0, 2(2-μ1-μ2)+2-μ1μ2lnR0>0, respectively, and both of them are obviously valid. Inequality (16c) may be expressed as(18)R0<expμ1+μ2μ1μ2.Hence, (x1∗,x2∗) 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 λ 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 R0 through an increase of F. Thus the bifurcation takes place at F=FT=μ2exp((μ1+μ2)/(μ1μ2)). Moreover, the eigenvalues may be expressed as(19)λ=122-μ1-μ2±41-μ11-μ2-μ1μ21-β1x2∗-2-μ1-μ22iand an easy computation shows that(20)ddFλ=12μ1e-μ1+μ2/μ1μ2>0(evaluated at F=FT) which proves that λ leaves the unit circle at threshold.
In order to show that the quasiperiodic orbit is stable when F>FT, F-FT 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(21)z⟶zeiθ01+F+dz2+Oz4(for details, cf. [18] or [19]) and the sign of a=Re(d) will determine the nature of bifurcation, a<0 implies supercritical, and a>0 implies subcritical.
Regarding (1) and (2) we may partly rest upon findings obtained in [13] and express a as(22)a=-β1216μ1+μ22μ1μ22+μ1+μ22μ1μ2-μ1+μ22-μ1μ2which is negative whenever β1>0, 0<μ1,μ2<1. 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 (x1∗,x2∗), R0<exp((μ1+μ2)/μ1μ2). Figure 1(b) shows the situation after the supercritical bifurcation R0>exp((μ1+μ2)/μ1μ2). 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 R0 interval but eventually chaos is introduced when the curve starts to break up. This is displayed in Figure 1(c).
(a) An orbit converging towards the stable equilibrium. Parameter values β1=2, β2=0, F=20, and μ1=μ2=0.5. (b) A stable attractor generated by (1) and (2). Parameter values as in (a) except F=30. (c) The invariant curve has broken up. Parameter values as in (a) and (b) except F=107.
(2) Next, assume β1=β2=β (same cannibalism pressure on newborns and older immature individuals).
In this case(23)x1∗,x2∗=μ2μ11βlnK,1βlnK,where K=μ1(R0-1)+1 and R0=F/μ2 and we notice that K′R0>0.
Stability criteria (16a) and (16b) reduce to μ2lnK>0, 2(1-μ2)+μ2lnK+2(1-μ1)K-1>0, respectively, and both of them clearly hold. Criterion (16c) may be written as(24)μ2lnK+1-μ1K<1+μ2.Obviously, R0→1+ implies that the left hand side of (24) is less than the right hand side but as R0 increases there must exists a threshold where a (supercritical) Neimark-Sacker bifurcation occurs. Thus, qualitatively, we have arrived at the same situation as in the β1≠0, β2=0 case.
Beyond instability we observe quasiperiodic orbits restricted to an invariant curve (cf. Figure 2(a)). As we continue to increase R0 the curve becomes kinked (Figure 2(b)) and therefore not topologically equivalent to a circle anymore. Eventually, the curve breaks up here too. However, note that we use the same F value when we generated Figures 1(c) and 2(b). Hence, one may argue that if both newborns and older immature individuals are exposed to cannibalism stability properties are improved compared to the case where β2=0.
(a) An invariant curve just beyond instability threshold. Parameter values β1=β2=2, μ1=μ2=0.5, and F=20. (b) The attractor when F=107. Other parameter values as in (a).
(3) The third case to discuss is β1=0, β2≠0 which biologically means that newborns are not exposed to cannibalism from older mature relatives but must reach a certain size (or age) to serve as food. From Theorem 1 we cannot conclude that (1) and (2) are a permanent population model. There is simply too little density dependence. The equilibrium point(25)x1∗,x2∗=μ2μ11β2ln1-μ11-μ1R0,1β2ln1-μ11-μ1R0exists and is stable only in the tiny interval 1<R0<1/μ1. If R0>1/μ1 the population goes to infinity. Note that such a phenomenon is by no means unique. It has been detected in other nonlinear models as well; see, for example, [20] or the prey-predator model in [8]. When R0>1/μ1 models (1) and (2) behave more or less as a linear model as accounted for in [21]. Also, compare [22].
4. Cannibalism and Harvest
Let us now focus on the role of harvest and the combined effect of cannibalism and harvest with respect to stability. We concentrate on cases where (1) and (2) are a permanent population model and we restrict the analysis to (i) h1=0, h2≠0, (ii) h1≠0, h2=0, and (iii) h1=h2=h≠0. Under these restrictions we have computed equilibria and stability conditions and the results are presented in Table 1. The various stability criteria are found from (16c). Our analysis shows that (16a) and (16b) always hold.
β/h
R0
(x1∗,x2∗)
Stability criterion
(A)
β1≠0β2=0h1=0h2≠0
1-h2F1-1-μ21-h2
1-1-μ21-h2μ11-h21β1lnR0,1β1lnR0
lnR0<μ1+μ2+1-μ2h2μ11-1-μ21-h2
(B)
β1≠0β2=0h1≠0h2=0
1-h1μ1Fμ21-1-μ11-h1
μ2μ11β1lnR0,1β1lnR0
lnR0<1+μ2-(1-μ1)(1-h1)μ21-1-μ11-h1
(C)
β1≠0β2=0h1=h2=h
1-h2μ1F1-1-μ11-h1-1-μ21-h
1-1-μ21-hμ11-h1β1lnR0,1β1lnR0
lnR0<2-1-h2-μ1-μ21-1-μ11-h1-1-μ21-h
(D)
β1=β2=βh1=0h2≠0
1-h2F1-1-μ21-h2
1-1-μ21-h2μ11-h21βlnKh2,1βlnKh2, where Kh2=μ1(R0-1)+1
1-1-μ21-h2lnKh2+1-μ1Kh2-1hh<2-1-μ21-h2
(E)
β1=β2=βh1≠0h2=0
1-h1μ1Fμ21-1-μ11-h1
μ2μ11βlnKh1,1βlnKh1, where Kh1=R0-1-μ11-h1R0-1
μ2lnKh1+1-μ11-h1Kh1<1+μ2
(F)
β1=β2=βh1=h2=h
1-h2μ1F1-1-μ11-h11-1-μ21-h2
1-1-μ21-hμ11-h1βlnKh,1βlnKh, where Kh=R0-1-μ11-hR0-1
1-1-μ21-hlnKh+1-μ11-hKh-1hh<2-1-μ21-h
We start by considering the case of β1≠0 and β2=0 which is covered by (A), (B), and (C) in Table 1.
If μ1<μ2 and h1=0 and h2≠0 numerical experiments show that the size of the total equilibrium population at instability threshold is larger than in case of no harvest. A natural way to implement this is to say that an increase of harvest of the mature population leads to better stability properties of the whole population (the size of the mature equilibrium population is reduced while the size of the immature population is increased). On the other hand, h1≠0 and h2=0 result in a smaller equilibrium population at instability threshold compared to the case of no harvest. Hence, harvest acts in a destabilizing fashion. If we apply the same harvest pressure h (h=h1=h2) on the subpopulations we experience that h acts as a weak destabilizing effect which means that the destabilizing effect of harvesting fractions of the immature population is somewhat stronger than the stabilizing effect of harvesting fractions of the mature population.
Turning to the case μ1=μ2 much of the same picture emerges. An increase of h2 stabilizes while an increase of h1 destabilizes. The difference between this case and μ1<μ2 is that the effect from harvesting the mature population now dominates. Thus, an increase of h turns out to be a (weak) stabilizing effect. Finally, if μ1>μ2 we find (as before) that an increase of h1 acts in destabilizing fashion and that an increase of h2 acts in stabilizing fashion. The combined effect of h1 and h2 acts in a weak destabilizing way. Hence, the overall conclusion from the cases discussed above is that an increase of h1 is a destabilizing effect while an increase of h2 tends to stabilize the dynamics. The effect of increasing h (h=h1=h2) depends on μ1 and μ2.
Let us now consider the cases where not only newborns but also immature relatives are exposed to cannibalism from the mature population. These cases are covered by (D), (E), and (F) in Table 1. Assuming μ1<μ2 we have by use of (D) adjusted F such that the dynamics in absence of harvest is restricted to an invariant curve (numerical example, μ1=0.3, μ2=0.7, and F=23). Now, keeping h2=0, an increase of h1 makes the invariant curve disappear and we observe a weakly attracting fixed point. The same qualitative picture emerges both when h1=0 and we increase h2 and also when h1=h2=h is increased. Thus, by use of (D), (E), and (F) and numerical experiments we conclude that harvest acts in a stabilizing fashion. Moreover, we experience the same situation under the assumption μ1=μ2. This is displayed in Figures 3(a) and 3(b). In Figure 3(a) (no harvest) map (1) generates an orbit restricted to an invariant curve. In Figure 3(b) where h2=0.25 we see a weakly stable fixed point together with some initial transients.
(a) An invariant curve generated by (1) and (2). Parameter values μ1=μ2=0.5, F=20, and h1=h2=0. (b) An attracting fixed point. Parameter values as in (a) but h2=0.25.
The final case to discuss is μ1>μ2 and again, in the absence of harvest, it is possible to obtain nonstationary dynamics provided F is sufficiently large. If h1=0 an enlargement of h2 leads to a “larger radius” of the invariant curve; hence we observe the same picture as earlier but the population is located further away from the fixed point. Consequently, it is natural to classify an increase of h2 as a destabilizing factor. On the other hand, an increase of h1 (h2=0) acts in stabilizing fashion in the same way as in the μ1<μ2 and μ1=μ2 cases. The combined effect of increasing both h1 and h2 turns out to be destabilizing (in a rather weak fashion).
5. Summary and Discussion
By use of a stage-structured population model, the purpose of this paper has been to reveal the dynamical consequences of cannibalism and the combined consequences of cannibalism and harvest. Despite the fact that there are two stages only (immature and mature) the parameter space is huge so some care should be taken with respect of drawing too strong conclusions.
Considering the effect of cannibalism alone, in the cases β1≠0, β2=0, or β=β1=β2, stability criteria are given by (18) and (24), respectively. Here we may notice that neither (18) nor (24) contains any cannibalism parameters which means that the values of βi contribute to the size of the equilibrium population only, not to qualitative changes of the dynamics. Nonstationary dynamics is achieved through an enlargement of R0 (or F). If only newborns are exposed to cannibalism it is possible to generate nonstationary as well as chaotic dynamics. When both βi and β2≠0 the same qualitative picture is established but this implies much larger R0 values. Thus a natural interpretation is to say that the more the immature population suffers from cannibalism, the better the stability properties of the population as a whole are.
Now, assuming that the population is exposed to both cannibalism and harvest, let us first comment on the situation where β1≠0, β2=0. Then the value of R0 and stability conditions are given by (A), (B), and (C) in Table 1 and in all cases (μ1<μ2, μ1=μ2, μ1>μ2) our finding is that an increase of harvest h1 of the immature part of the population acts in a destabilizing fashion while an enlargement of h2 improves stability properties. The effect which is the most profound one depends on μ1, μ2.
If both newborns and older immature relatives are exposed to cannibalism (β=β1=β2≠0) stability criteria and R0 are given by (D), (E), and (F) in Table 1. As is shown, the results obtained here are in many respects quite different from the case β2=0. Now, μ1≤μ2 implies an increase of both h1 and h2 that act as stabilizing factors. If μ1>μ2 then an increase of h2 turns out to improve stability properties. Thus, what these results show is that the impact of harvest may act differently depending on which part of the immature population is exposed to cannibalism.
It is tempting to compare findings above with results obtained from analysis of age-structured Leslie matrix population models, that is, models on the form(26)x1,t+1=∑i=1nFe-xtxi,t1-h1,x2,t+1=P1x1,t1-h2,⋮xn,t+1=Pn-1xn-1,t1-hn.Model (26) consists of n nonoverlapping age classes x1,…,xn and xt=∑xi is the total population. Moreover, F and hi have the same meaning as in (1) and Pi, 0<Pi≤1, are the year to year survival probabilities. Observe that, in case of n large, individuals spend most of their lives as grownups. In particular this is the case when the year to year survival probabilities are close to unity. This corresponds to μ1>μ2 in (1). (Indeed, if μ1 is large, then individuals spend only a tiny part of their lives as juveniles and if μ2 is small then the “grownup part” is large.) Note that (1) and (26) rest upon somewhat different prerequisites as accounted for in [22].
Now, considering (26), in the absence of harvest we find that an increase of the number of age classes turns out to be a stabilizing effect (cf. [23]). Harvest acts more and more in destabilizing fashion as n is increased, especially when the survival probabilities become large. Hence, in the interplay between increasing the number of age classes and harvest the destabilizing effect of the latter turns out to be the dominant one. Thus, in many respects, this is consistent with the qualitative findings from (1) in the μ1>μ2 case.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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