Complex dynamics of survival and extinction in simple population models with harvesting

Abstract

We study the effects of constant harvesting in a discrete population model that includes density-independent survivorship of adults in a population with overcompensating density dependence. The interaction between the survival parameter and other parameters of the model (harvesting rate, natural growth rate) reveal new phenomena of survival and extinction. The main differences with the dynamics of survival and extinction reported for semelparous populations with overcompensatory density dependence are that there can be multiple windows of extinction and conditional persistence as harvesting increases or the intrinsic growth rate is increased, and that, in case of bistability, the basin of attraction of the nontrivial attractor may consist of an arbitrary number of disjoint connected components.

Appendices

Appendix A

This appendix is devoted to give more details on the derivation of Eq. 2 from ecological models for the Ricker and the logistic maps. An interesting derivation of Eq. 2, with f being the Ricker map, is given in the book of Thieme (2003). Taking into account the survivorship assumption, the resulting difference equation is

$$ y_{n+1}=y_n\left(q+\gamma e^{-y_n} \right), $$

(6)

where q ∈ [0,1] is an adult’s probability of surviving 1 year including the reproductive season, and γ is the number of per capita offspring still alive after 1 year if there is no cannibalism (we refer to Thieme (2003, Section 9.2) for more details).

If q + γ > 1, then there is a unique positive equilibrium K of Eq. 6. Notice that this provides a dependence relation among the parameters q, K, and γ, namely,

$$ 1=q+\gamma e^{-K}. $$

(7)

This relation is sometimes referred to as the balance equation (May 1980).

Replacing Eq. 7 into Eq. 6 leads to

$$ y_{n+1}=y_n\left(q+(1-q)e^{K-y_n}\right)=qy_n+(1-q)y_ne^{K-y_n}. $$

Setting y _n  = Kx _n , r = K, q = α, the positive equilibrium is normalized to 1, and Eq. 6, reads

$$ x_{n+1}=\alpha x_n+(1-\alpha)x_ne^{r(1-x_n)}. $$

Using similar arguments, May (1980) (see also Fisher 1984) derived equation

$$ y_{n+1}=\alpha y_n+(1-\alpha)y_n\left(1+q\left(1-\frac{y_n}{K}\right)\right)^z, $$

(8)

used by the International Whaling Commission (IWC) (Beddington 1978; Beddington and May 1980) to describe the dynamical behavior of baleen whale populations. The meaning of the parameters in Eq. 8 is the following: K is the positive equilibrium, q is the maximum increase in fecundity of which population is capable at low densities, and z measures the severity with which the density-dependent response is manifested. The case z = 1 corresponds to the logistic assumption, in which the density-dependent increase in fecundity is linear. After normalization, Eq. 8 with z = 1 is rewritten as

$$ x_{n+1}=\alpha x_n+(1-\alpha)rx_n(1-x_n), $$

with r = 1 + q, qy _n  = (1 + q)Kx _n .

Appendix B

Assume that f satisfies A1–A4, and lim_x→ ∞ f(x) = 0. Since f′(c) = 0, f′(x) < 0 for all x > c, and lim_x→ ∞ f(x) = 0, it follows that f has an inflexion point at f(δ), for some δ > c. Moreover, this is the unique inflexion point of f. Indeed, condition A3 ensures that f has, at most, one inflexion point on each interval not containing a critical point (this is a consequence of Lemma 3 in Schreiber 2001). On the other hand, condition A4 prevents the existence of inflexion points on [0,c]. Therefore, it is clear that f′(δ) =  min {f′(x) : x > 0}.

Consider F _α (x) = αx + (1 − α)f(x), with α ∈ (0,1). Since $F_{\alpha}'(x)=0\Longleftrightarrow f'(x)=-\alpha/(1-\alpha)$, we distinguish two cases:

• (a) Monotone case. If f′(δ) ≥ − α/(1 − α), then

  $$ F_{\alpha}'(x)\!=\!\alpha\!+\!(1\!-\!\alpha)f'(x)\!\geq\! \alpha\!+\!(1\!-\!\alpha)f'(\delta)\!\geq\! 0,\,\forall\, x\!>\!0. $$

  Hence, F _α is nondecreasing on (0, ∞ ), and therefore, there are only two possible modes of survival/extinction for Eq. 4:

  • Extinction: if 0 is the unique fixed point of F _α,d, then $\lim_{n\to\infty}F_{\alpha,d}^n(x)=0$ for all x > 0.

  • Bistability: if there are 0 < K ₁ ≤ K ₂ such that F _α,d(K ₁) = K ₁, F _α,d(K ₂) = K ₂, then $\lim_{n\to\infty}F_{\alpha,d}^n(x)=0$ for all x ∈ (0, K ₁) and $\lim_{n\to\infty}F_{\alpha,d}^n(x)=K_2$ for all x > K ₁.

• (b) Bimodal case. If f′(δ) < − α/(1 − α), then equation F _α ′(x) = 0 has two solutions c₁, c₂, with 0 < c₁ < δ < c₂. Moreover, F _α ′(x) > 0 on (0,c₁) ∪ (c₂, ∞ ), and F _α ′(x) < 0 on (c₁,c₂). This means that F _α (c₁) is a local maximum, and F _α (c₂) is a local minimum.

For example, for the Ricker map $f(x)=x\exp(r(1-x))$, the inflexion point is reached at δ = 2/r, and the minimum of f′ is $f'(\delta)=-\exp(r-2)$. Thus, $F_{\alpha}(x)=\alpha x+(1-\alpha)x\exp(r(1-x))$ is bimodal for all r > 2 + ln (α/(1 − α)).

Appendix C

We recall (Block and Coppel 1992, Section III.3) that, if g is a continuous map from a real interval I into itself, a point y is homoclinic to a periodic point z of period k ≥ 1 if f ^k·n(y) = z for some n > 0, and y belongs to the unstable manifold of z. As proved in (Proposition 21, p. 64, of the same book), a continuous map is chaotic if and only if it has a homoclinic point.

It is clear that the condition for chaotic semi-stability in Theorem 1 of Schreiber (2001) is equivalent to saying that the critical point c is homoclinic to the least positive fixed point of f. The orbit formed by the homoclinic point, its preimages, and its (finite) forward orbit is a homoclinic orbit. Moreover, if this orbit contains a critical point, it is called a degenerate homoclinic orbit (Devaney 1989, Section 1.16). See Fig. 10, where a degenerate homoclinic orbit is represented.

Fig. 10

A degenerate homoclinic orbit to a fixed point z. For the critical point y, g ²(y) = z, and the preimages g ^− n(y) converge to z as n→ ∞

For α and d fixed, consider the map $F_{\alpha,d}(x)=\max\{\alpha x +(1-\alpha) x \exp(r(1-x))-d,0\}$ used in the paper as a paradigm of the influence of harvesting in a population model governed by a bimodal map.

For α = 0, the catastrophe bifurcation (Fig. 6a) occurs when the basin of attraction of the chaotic attractor collides with the unstable fixed point arising in the tangent bifurcation, that is, when $F_{0,d}^2(c_1)=K_1$. At this bifurcation point, the orbit of c ₁ is homoclinic to K ₁, as shown in Fig. 10.

For α > 0, there are more modes of survival and extinction. They are typically created, as r is increased, in tangent bifurcations for an iterate $F_{\alpha,d}^k$, k ≥ 1, and destroyed when the basin of attraction of the chaotic attractor created by a period doubling cascade from the stable k-cycle collides with the unstable k-cycle originated in the tangent bifurcation. When k = 1, this simply means that $F_{\alpha,d}^{2}(c_1)=K_1$. For k > 1, there is a degenerate homoclinic orbit to the unstable k-cycle. In particular, this remark provides the formula $F_{\alpha,d}^{k+1}(c_1)=F_{\alpha,d}^{2k+1}(c_1)$, which is useful to numerically determine the bifurcation point.

To explain how the survival mode is destroyed in a crisis bifurcation, let us look at the example given in the section “New modes of survival and extinction,” that is, $F_{\alpha,d}(x)=\max\{0.15 x +(1-0.15) x \exp(r(1-x))-0.4,0\}$. As indicated in the section “New modes of survival and extinction,” a transition from essential extinction to bistability takes place as the parameter r is increased, via a tangent bifurcation for the second iteration $F_{\alpha,d}^2$.

It is useful to use a graphic representation of $F_{\alpha,d}^2$. In Fig. 11, this is made for the value r = 4.6, belonging to the region of bistability. After the tangent bifurcation, two 2-cycles arise. The unstable 2-cycle {x _a ,x _b } gives two fixed points of $F_{\alpha,d}^2$. Now, we can determine two points x _c , x _d , such that

$$\begin{array}{rll} x_c&=&\min\left\{x>x_a\, :\, F_{\alpha,d}^2(x)=x_a\right\}\; ;\\ x_d&=&\min\left\{x>x_b\, :\, F_{\alpha,d}^2(x)=x_b\right\}. \end{array}$$

Fig. 11

Graphic representation of the map $F_{\alpha,d}^2=F_{\alpha,d}\circ F_{\alpha,d}$, where $F_{\alpha,d}(x)=\max\{0.15 x +(1-0.15) x \exp(r(1-x))-0.4,0\}$ with r = 4.6

Denote I ₁ = [x _a ,x _c ] and I ₂ = [x _b ,x _d ]. It is clear that F _α,d(I ₁) = I ₂ and F _α,d(I ₂) = I ₁. Moreover, the set of initial conditions for which the orbit persists indefinitely is I ₁ ∪ I ₂. The crisis bifurcation occurs when y ₁: = F _α,d(c ₁) = x _d , that is, when $y_3:=F_{\alpha,d}^3(c_1)=x_b$. Notice that this relation defines the homoclinic orbit from c ₁ to the cycle of period two. Moreover, since $F_{\alpha,d}^2(x_b)=x_b$, we get the formula $F_{\alpha,d}^3(c_1)=F_{\alpha,d}^5(c_1)$, from which the bifurcation value r = 4.6488 is determined numerically. In Fig. 12, we represent the map F _α,d with r = 4.6488, for which $F_{\alpha,d}^3(c_1)=x_b,$ giving place to a degenerate homoclinic orbit.

Fig. 12

Graphic of the map $F_{\alpha,d}(x)=\max\{0.15 x +(1-0.15) x \exp(r(1-x))-0.4,0\}$ with r = 4.6488. There is a homoclinic orbit to the 2-cycle {x _a ,x _b }