Chaos control in discrete population models (Harvesting and Dynamics)

Transcription

1 Ricker Clark Chaos control in discrete population s ( ) Departamento de Matemática Aplicada II Universidad de Vigo, Spain!!! "#$!%& "#! '("$)(*"'+(*,!-+(.$)$(-$! June 3, (!/'..$)$(-$!$01*"'+(2! *(/!*33,'-*"'+(2!4'-/$*56%78!!! 9:;<=>?+@=A!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!9=B!5C?76D!56%7!! "#$! %&'()*+*)'!,%--*..$$! %/!.#$! 01.#! 2).$&)(.*%)(3! 4%)/$&$),$!%)!5*//$&$),$!678(.*%)9!():!;<<3*,(.*%)9!*)=*.$9!! 2-'$("'.'-!-+99'""$$! Z*):(!;33$)K!"$R(9!"$,#!D)*=$&9*.>K!!DB;!

2 Single-species population s MARINE ECOLOGY PROGRESS SERIES Vol. 274: (2004) Ricker Clark Pamela M. Mace (Ministry of Fisheries, New Zeal) In defence of fisheries scientists, single-species s other scapegoats: confronting the real problems" The much reiterated claim that single-species fisheries assessment s have failed need to be supplanted with ecosystem s distorts the facts... Given the record of fisheries management to date, it is difficult to underst the reasoning that leads to the conclusion that ecosystem-based approaches will succeed where single-species approaches have failed... Unlike most ecosystem objectives, the goals of single-species management are usually easily understood, straightforward, operational. Even by itself, successful single-species management is likely to go a long way towards achieving many so-called ecosystem objectives.

3 Single-species population s Strategy of constant effort harvesting Ricker Clark We consider a harvesting assuming a strategy of constant effort, that is, the catch is proportional to the population size. Our aim is discussing some interesting phenomena that arise in simple s, the role of different parameters, such as harvesting effort, census timing, harvest timing; we shall restrict our study to one-dimensional difference equations, choosing as a prototype the classic Ricker x n+1 = x n e r(1 xn), n = 0, 1,..., two of its variants that allow for the possibility of adult survivorship Allee effects, respectively. This talk is based on recent work developed with some colleagues, especially Alfonso Ruiz Herrera (Universidad de Granada, Spain) Frank Hilker (University of Bath, UK)

4 Single-species population s The Ricker some variants 2.5 Ricker Clark (I) (II) (III) (I) Original Ricker map f (x) = x e r(1 x), r > 0. (II) Clark map f (x) = αx + (1 α)x e r(1 x), α (0, 1). (III) map f (x) = βx 1 + βx x er(1 x), β > 0.

5 Ricker Clark The Ricker

6 The original Ricker Constant effort harvesting Ricker Clark The strategy of constant effort harvesting assumes that a percentage γ x of the population is removed. Effects of harvesting: x n+1 = (1 γ)x n e r(1 xn), γ (0, 1). Reduces complexity (stabilization); Reduces variability (increases constancy); Overharvesting leads to extinction population size, x 2.5 r=3 harvesting effort, γ

7 after or before reproduction? does it matter? Ricker Clark The simplest harvesting assumes only two processes: reproduction harvesting. Denote by f (x) = x e r(1 x), r > 0, the recruitment function h(x) = (1 γ)x, γ (0, 1) the harvesting action. x n h(x n ) Recruitment x n+1 = f (h(x n )) From a mathematical point of view, since h is a homeomorphism, the discrete dynamical systems generated by f h h f are topologically conjugated. Hence, for the usual Ricker, the dynamical consequences of harvesting do not depend on the harvesting time. + Reproduction x n+1 = f (h(x n )) Reproduction + x n+1 = h(f (x n ))

8 after or before reproduction? What we observe is not the same: the Hydra effect 2.5 Parent Stock (h(f(x)) 2.5 Recruits (f(h(x)) Ricker Clark population size, x harvesting effort, γ population size, x harvesting effort, γ Census timing does really matter. Assume we only measure population once a year (just after harvesting or just after reproduction). In other words, this means that we are measuring either parent stock or recruits, respectively. There is a quantitative disparity in the s: the that census recruits exhibits the so-called Hydra effect, popularized after the survey of Peter Abrams (Ecol. Letters, 2009). Previously it have been termed overcompensation or paradoxical increase.

9 Population equilibrium (K) The hydra effect x n+1 = f ((1 γ)x n) (1) ; x n+1 = (1 γ)f (x n) (2) Ricker Clark The hydra effect is a consequence of the overcompensatory character of the Ricker map, it is easy to explain through the geometric interpretation of the positive equilibrium. Notice that for the Ricker the positive equilibrium equals the average population size even in the case of instability. K γ is an equilibrium of (2) iff f (K γ ) = K γ /(1 γ), K γ = f (K γ ) is the corrresponding equilibrium of (1). Notice the risk of censusing just after reproduction (fast decline to extinction). y y=f(x) y=x/(1-γ) γ Census after reproduction Kγ K 0 y=x γ=0 1 Kγ Kγ Census after harvest 0 K γ K 0 x 0 effort (γ) γ*

10 Having intervention time into account Seno s Ricker Clark Hiromi Seno (Math. Biosci., 2008) suggested a in which it is assumed that there is a specific season of length T during which the individual accumulates the energy for reproduction. The cumulative density effect during the specific season determines the recruitment. can take place at any moment τ = θt, 0 θ 1, during the specific season. The case θ = 0 corresponds to harvesting before the specific season, θ = 1 to harvesting after the specific season. harvesting reproduction harvesting reproduction x n τ=0 τ=θt τ=t specific season x n+1 τ=0 τ=θt τ=t specific season

11 Having intervention time into account Seno s harvesting reproduction harvesting reproduction Ricker Clark x n τ=0 τ=θt τ=t specific season x n+1 τ=0 τ=θt τ=t specific season If h is the harvesting map f is the recruitment function, Seno s writes x n+1 = θh(f (x n )) + (1 θ)f (h(x n )). Seno showed that the hydra effect occurs for low values of θ, that is, the earlier we harvest, the most the population size is increased (a possible biological interpretation is that harvesting releases part of the population from intraspecific competition.)

12 Intervention time dynamics x n+1 = θh(f (x n)) + (1 θ)f (h(x n)). Ricker Clark We point out that Seno s map x θh(f (x)) + (1 θ)f (h(x)) is a convex combination of the maps h f f h which represent harvesting after reproduction (θ = 1) harvesting before reproduction (θ = 0), respectively. We know that the cases θ = 0 θ = 1 are dynamically equivalent. However, in general Seno s map is not conjugated to h f f h for an intermediate value θ (0, 1). Thus, a question naturally arises: How does intervention time affect the qualitative behavior of the harvesting?

13 Intervention time dynamics x n+1 = θh(f (x n)) + (1 θ)f (h(x n)), f (x) = xe r(1 x), h(x) = (1 γ)x, (S) Ricker Clark Effects of intervention: It does not change the critical value γ leading to extinction; Average population size is a decreasing function of θ; Intermediates values of θ reduce complexity. intervention time, θ Oscillations Stability harvesting effort, γ γ 1 Theorem If the positive equilibrium of (S) is asymptotically stable for θ = 0 ( θ = 1) then the positive equilibrium of (S) is asymptotically stable for all θ (0, 1).

14 Intervention time dynamics x n+1 = θh(f (x n)) + (1 θ)f (h(x n)), f (x) = xe r(1 x), h(x) = (1 γ)x, (S) Ricker Clark The stabilization properties r=4, γ=0.7 5 may be very helpful for 4 management. Assume that we plan to catch 70% from 3 the stock. For r = 4, θ = 0, 2 equation (S) is chaotic, inducing a risk of stochastic 1 extinction because the 0 population floor is very low θ1 0.7 θ2 population size, x intervention time, θ If we want to prevent the population to reach low densities, an optimal intervention time would be θ = θ If we aim to control a plague avoiding bursts of high density, a good intervention time would be about θ = θ

15 Ricker Clark The Clark

16 Ricker with adult survivorship x n+1 = αx n + (1 α)f (x n) := F α(x n), α (0, 1), f (x) = xe r(1 x) (R C) Ricker Clark Parents Pn Juveniles Jn Survivorship Recruits f(p n) Parents Pn+1 Recruitment Equation (R-C) was introduced by C. W. Clark (1976), it represents the simplest way to consider age structure in a discrete of population dynamics. x n is the adult population, α (0, 1) is the survivorship rate of adults f is the stock-recruitment function. The parameter α can be interpreted as the fraction of energy invested into adult survivorship rather than reproduction.

17 can magnify fluctuations of population abundance x n+1 = α(1 γ)x n + (1 α)f ((1 γ)x n) ; f (x) = xe r(1 x) (R C) Ricker Clark Recent studies indicate that greater mortality does not always tend to simplify the dynamics. For example, C. N. K. Anderson et al. (Nature, 2008) A. O. Shelton M. Mangel (PNAS, 2011) show that fishing can magnify fluctuations in population abundance. E. F. Zipkin et al. (Ecological Applications, 2009) state, based on numerical observations, that instability as a result of constant effort harvesting occurs when both maximum per capita recruitment adult survivorship are high. It is not difficult to prove that the usual (semelparous) Ricker cannot explain this phenomenon. We study this problem in the Clark (R-C) using the concept of bubbling.

18 Bubbling A formal definition of a bubble Ricker Clark Consider a family f λ : I I of C 1 -maps defined on a real interval I depending smoothly on a parameter λ. We assume that, for each value of λ, the ω-limit set ω λ (x 0 ) of any initial condition x 0 I is invariant, nonempty compact. Assume also that there is an interval J λ 1, λ 2 J such that f λ has a continuous branch of equilibria K λ satisfying f λ (K λ) > 1 for all λ (λ 1, λ 2 ), f λ (K λ) < 1 for all λ J \ (λ 1, λ 2 ). Definition In the above conditions, we say that the family {f λ } λ J exhibits bubbling if, for some x 0 I, ω λ (x 0 ) = K λ for all λ J \ (λ 1, λ 2 ), M(λ, x 0 ) := max ω λ (x 0 ) > min ω λ (x 0 ) := m(λ, x 0 ) for all λ (λ 1, λ 2 ). In this case, we refer to the following set as a bubble: B = (λ, m(λ, x 0 )) (λ, M(λ, x 0 )) λ 1 <λ<λ 2 λ 1 <λ<λ 2

19 A simple bubble x n+1 = α(1 γ)x n + (1 α)f ((1 γ)x n) ; f (x) = xe r(1 x) (R C) The simplest bubble results from a period doubling bifurcation followed by a period-halving bifurcation. Ricker Clark population size, x 2.5 α=5, r= harvesting effort, γ A bubble in the bifurcation diagram of (R-C) with r = 4 α = 5. As far as we know this is the first time that a bubble was found in a one-dimensional discrete population. However, they appeared previously in other s such as the famous three-dimensional map of the flour beetle (J. M. Cushing et al., Chaos in Ecology", 2003).

20 Time series for r = 4, α = 5 x n+1 = α(1 γ)x n + (1 α)f ((1 γ)x n) ; f (x) = xe r(1 x) (R C) Ricker Clark population size, x 2.5 α=5, γ=0 No harvesting: stable equilibrium generations, n population size, x α=5, γ= % harvesting rate: sustained oscillations generations, n 2.5 α=5, γ= α=5, γ=0.96 population size, x Increasing harvesting: stabilization population size, x Overharvesting: extinction generations, n generations, n

21 More complex bubbles x n+1 = α(1 γ)x n + (1 α)f ((1 γ)x n) ; f (x) = xe r(1 x) (R C) The dynamics inside a bubble can be very simple or even chaotic. Ricker Clark population size, x 7 α=0.73, r= population size, x α=0.675, r= harvesting effort, γ harvesting effort, γ Bubbling scenario in equation (R-C) with r = 6. Left: nested period-doubling period-halving bifurcations for α = Right: increasing harvesting can lead to chaos inside the bubble for α =

22 Chaotic bubbles: paired cascades x n+1 = α(1 γ)x n + (1 α)f ((1 γ)x n), f (x) = xe r(1 x) (R-C) Ricker Clark Chaotic Bubbles" were recently studied by E. Ser J. Yorke in a series of papers ( ). They call these structures period-doubling paired cascades. population size, x α=2, r=4 Saddle-Node harvesting effort, γ Detail of a paired cascade for the (R-C) with r = 4.

23 Existence of bubbles x n+1 = α(1 γ)x n + (1 α)f ((1 γ)x n), f (x) = xe r(1 x) (R-C) Ricker Clark Theorem (A. Ruiz-Herrera, EL) Equation (R-C) exhibits bubbling as γ is increased if only if r > 3 α 1 (r) := 1 2 r < α < er e r 3 := α 2(r). survivorship rate, α Bubbles fecundity rate, r

24 Existence of bubbles x n+1 = α(1 γ)x n + (1 α)f ((1 γ)x n), f (x) = xe r(1 x) (R-C) Ricker Clark The previous result allows us to represent in the parameter plane (α, γ) the values for which bubbles appear in Equation (R-C) as γ is increased. survivorship rate, α r=4 Globally Stable Equilibrium Unstable Equilibrium Extinction harvesting effort, γ Regions of stability bubbling in the parameter plane (γ, α) for equation (R-C) with r = 4. There are bubbles for α (, 76).

25 Preventing bubbles x n+1 = θ(1 γ)f α(x n)+(1 θ)f α((1 γ)x n), F α(x) = αx +(1 α)f (x) (S) Ricker Clark Since harvesting induces some undesirable dynamical effects both in cases θ = 0 θ = 1 (recall that they are conjugated), an interesting problem consists of finding out whether it is possible to avoid such effects choosing an appropriate harvesting time during the specific season. We use again Seno s. population size, x 2.5 α=5, r=4 θ=0 θ= θ= harvesting effort, γ Bubbles for Seno s adult survivorship (S) with f (x) = x e 4(1 x), α = 5, different harvest times θ. The blue bubble corresponds to θ = 0, the red one to θ = 1. For θ = the positive equilibrium is globally stable for all values of γ, so harvesting does not destabilize it.

26 Stability Stability Preventing bubbles x n+1 = θ(1 γ)f α(x n) + (1 θ)f α((1 γ)x n) (S) Ricker Clark The (γ, θ) stability diagram shows that a suitable intervention time can prevent instabilities due to harvesting in the Clark. We show the diagram for the Clark map with r = 4, α = Oscillations θ2 2.5 (b) intervention time, θ Stability θ1 population size, x γ= Oscillations γ 1 harvesting effort, γ θ intervention time, θ θ

27 Preventing chaos extinction x n+1 = θ(1 γ)f α(x n) + (1 θ)f α((1 γ)x n) (S) Ricker Clark For low survivorship rates, the Clark can exhibit chaos. For example, for r = 4, α = 2, γ = 0.65, equation (S) with θ = 0 is chaotic. The bifurcation diagram shows that for an appropriate harvesting timing, equation (S) has a 2-periodic attractor that allows to increase the population floor; even if the mean population size is lower, this dynamical behavior reduces the chance of extinction due to stochastic perturbations. population size, x intervention time, θ Bifurcation diagram of (S) with r = 4, α = 2, γ = 0.65, using the intervention time θ as the bifurcation parameter. The red line represents the mean population size.

28 Ricker Clark The

29 Allee effects ovecompensation The Ricker Clark We finally consider the x n+1 = 2.5 βx 2 n 1 + βx n e r(1 xn), β, r > 0. The positive density factor I(x) = βx/(1 + βx) has been used by S. J. (2003) to mate limitation. The parameter β represents an individual s efficiency to find a mate.

30 Extinction Unusual dynamics of extinction x n+1 = (1 γ)(βx 2 n /(1 + βx n))f (x n), f (x) = xe r(1 x) (R-S) Ricker Clark population size, x 2.5 Essential extinction Collapse due to overproduction Bistability Collapse due to overharvesting harvesting effort, γ Bifurcation diagram of the (R-S) with r = 4, β = 4. There is an unusual behaviour of extinction: populations can persist within a b of high depletion whereas extinction occurs for lower depletion rates. This phenomenon is typical in a strategy of constant quota harvesting, has been uncovered by S. Sinha S. Parthasarathy (1996). For constant effort harvesting in s with Allee effects, it was demonstrated by (2003). The first collapse is due to a boundary collision.

31 Influence of harvest timing x n+1 = θ(1 γ)f β (x n) + (1 θ)f β ((1 γ)x n) F β (x) = (βx 2 /(1 + βx))xe r(1 x) (R S) Ricker Clark Generic possibilities for the dynamics of (R-S): (monostable) extinction due to overharvesting; bistability between extinction (possibly complex) survival; essential extinction. Theorem Denote by γθ the critical value of the harvesting time for a saddle node bifurcation to occur in (R-S). Then γθ < γ 1 = γ 0, for all θ (0, 1). This means that intermediate values of θ make the population more prone to extinction by overharvesting.

32 Influence of harvest timing x n+1 = θ(1 γ)f β (x n) + (1 θ)f β ((1 γ)x n) F β (x) = (βx 2 /(1 + βx))xe r(1 x) (R S) Ricker Clark x θ γ= x θ γ= x γ=0.866 x γ= Bistability Extinction Bistability θ θ Equilibria of (R-S) with r = 4, β = 4. We show why intermediate harvesting times can induce extinction. Although both the carrying capacity K 1 the Allee threshold K 2 decline with later harvesting times, they do so with different rates. If we increase the harvesting moment θ, K 1 K 2 approach each other. If we then increase the harvesting effort, the two equilibria collide this results in extinction.

33 Influence of harvest timing x n+1 = θ(1 γ)f β (x n) + (1 θ)f β ((1 γ)x n) F β (x) = (βx 2 /(1 + βx))xe r(1 x) (R S) Ricker Clark harvesting effort, γ Bistability Essential extinction intervention time, θ population size, x 2.5 γ=9 Essential extinction Essential extinction intervention time, θ Bifurcation diagrams of (R-S) with r = 4, β = 4. In contrast with the previous observation for high depletion rates, for low harvesting rates intermediate values of the harvesting moment θ may prevent essential extinction.

34 Influence of harvest timing x n+1 = θ(1 γ)f β (x n) + (1 θ)f β ((1 γ)x n) F β (x) = (βx 2 /(1 + βx))xe r(1 x) (R S) Ricker Clark population size, x γ= intervention time, θ Bifurcation diagram of (R-S) with r = 4, β = 4. As in the previous s, for moderate harvesting rates, intermediate values of the harvesting moment θ may help to stabilization.

35 Unexpected (or not) effects of harvesting Review comments Ricker Clark Regarding the complexity of the dynamics, populations with adult survivorship can be either stabilized or destabilized by harvesting. Typical mechanisms are period-halving bifurcations bubbling. In regard to population abundance, the most interesting phenomenon is the Hydra effect: population abundance can increase in response to an increase in its per-capita mortality rate. This phenomenon underlines the importance of census timing. Hydra effects are not exclusive of proportional harvesting, but also appear in strategies of constant harvesting conditional harvesting. In s with Allee effect, overharvesting may lead the population to extinction after a sudden collapse, but low harvesting rates may help the population to survive, preventing essential extinction due to overproduction.

36 Effects of harvest timing Review comments Ricker Clark In compensatory s, harvest time does not affect overharvesting leading to extinction, but harvesting at an intermediate moment of the season can reduce complexity, preventing chaos sometimes stabilizing the positive equilibrium. In s with Allee effect, intermediate harvesting timing can enhance both persistence as well as extinction; the actual outcome depends on the magnitude of the harvesting effort.

37 Final conclusions Consequences for management Ricker Clark To conclude, we emphasize that in designing management programmes of exploited populations, it is essential: A good knowledge of the that better fits the population growth: compensatory/ overcompensatory/ depensatory?... semelparous/ iteroparous? Census timing: How many times when do we measure? Harvest timing: As stated by H. Kokko (Wildlife Biol., 2001): Timing of harvesting may profoundly influence the impact on the population." We have used Seno s to allow for variable harvest times investigate its impact on the population dynamics.

38 I Ricker Clark E. L., Alfonso Ruiz-Herrera (2012) The hydra effect, bubbles, chaos in a simple discrete population with constant effort harvesting. Journal of Mathematical Biology 65, Frank Hilker, E. L. (2013), census timing hidden" hydra effects. Ecological Complexity 14, Begoña Cid, Frank Hilker, E. L. (2013) Harvest timing its population dynamic consequences in a discrete single-species. Submitted.