A pulse fishery model with closures as function of the catch: Conditions for sustainability
aa r X i v : . [ q - b i o . O T ] O c t A PULSE FISHERY MODEL WITH CLOSURES AS FUNCTIONOF THE CATCH: CONDITIONS FOR SUSTAINABILITY
FERNANDO C ´ORDOVA–LEPE, RODRIGO DEL VALLE, AND GONZALO ROBLEDO
Abstract.
We present a model of single species fishery which alternates closedseasons with pulse captures. The novelty is that the length of a closed seasonis determined by the stock size of the last capture. The process is described bya new type of impulsive differential equations recently introduced. The mainresult is a fishing effort threshold which determines either the sustainability ofthe fishery or the extinction of the resource. Introduction
Preliminaries.
This work proposes a model of management of closed seasons (also named seasonal closures ) for fisheries. The FAO’s fisheries glossary definesa closed season of a single marine species as the banning of fishing activity ( in anarea or of an entire fishery ) for a few weeks or months, usually to protect juvenilesor spawners . The closed seasons are implemented by a regulator authority in orderto limit the productive factors in certain area during a specific time interval.A fishery process could envisage alternated periods of closures and open seasons ,where the fishing is allowed. The closures are introduced according to bioeconomicalnecessities and its scope considers several spatio-temporal possibilities, leading toconcepts as: seasonal closures–no area closure , short term area closure–no seasonalclosure , short term time and fully–protected area , see e.g. , [13] for details.This article will consider the special –but important– case of bioeconomic pro-cesses having the following property [31]: low frequency / short duration of openseasons combined with large magnitude of capture . In this framework, we propose afeedback regulatory policy which defines the lenght of the next closure as a functionof the present captured biomass. A consequence of this regulation is that undercertain fishing effort (this concept will be explained later) threshold, the conver-gence towards a constant length closures ensuring the ecological sustaintability ofthe resource is obtained.The literature shows many cases of a long–term time and area closures. Anexample is the pacific herring ( Clupea pacificus ) fishery, where the regulatory mea-sures have included very short open seasons as two hours joined with other inputsand outputs restrictions [23]. Another example is given by a village–base manage-ment program in Vanuatu, where the stocks should be harvested in a sequence ofbrief openings interspresed with several years closures (see [18], [19] for details).Cases of annual fisheries combining short periods with large captures, followed by
Date : October 2011.
Key words and phrases.
Fisheries management, Impulsive differential equations, Stability,Sustainability. Captured Biomass years t on s Figure 1.
Captures of anchovy and common sardine (1991–1995)in southern Chilean coast [11]. The period with the largest fishingmortality is the summer, while the lowest one corresponds to theclosed season.
Captured Biomass years t on s Figure 2.
Idealized scheme of the previous figure: intense fishingseason alternated by absolute closures of nine months. If the lenghtof openings are small compared with closures and the harvestedstock is large, it can be considered as a pulse.(comparatively) low ones at the rest, are registered. For instance, the fishery ofcommon sardine (
Strangomera bentincki ) and anchovy (
Engraulis ringens ) in thesouthern Chilean coast has a behavior described by the Figure 1.The problem of determining the lenght of a closure (in a context of short–termopen seasons with intense fishing effort) ensuring the bioeconomical sustantabilityis a complex issue: indeed, short term closures followed by intense captures could induce an overexploitation of the resource. On the other hand, long–term closurescould have some unexpected drawbacks as: i)
Bio–sustainable economic rent withnegative average [6]. ii) Promotion of negative indirect effects [3], [17] when theresource is a top predator. An example is given by the closure (1989–1992) of themollusk
Concholepas concholepas in the chilean coasts [27]. iii) The phenomenonknown as race for fish : fishing units try to outdo one another in fishing power andefficiency during the brief openings [7].1.2.
Mathematical modeling.
In this paper, we will assume that the fisheryprocess has two different time–scales. In the first scale (closed seasons), the growthof a single unstructured marine resource is described by using ordinary differentialequations. Nevertheless, as the capture has a duration considerably shorter than theclosures lenght, every open season will be considered as an instant of capture, i.e. ,a term of a sequence { t k } k , this is the second scale (See Figure 2). In consequence,the global process will be described by an impulsive differential equation, IDE (see e.g. , [16], [24] for details).IDE equations have been used in the mathematical modeling of processes involv-ing impulsive harvesting: e.g. , [2], [4], [5], [14], [32], [33], [34], [35] (see [22] and [31]for more applications of IDE to bioeconomics and ecological processes respectively).In all these references, the following sequence of harvest instants is employed:(1.1) t k = kτ with k = 0 , , . . . , and τ > , which implies that τ = t k +1 − t k is the time between two consecutive captures.This paper follows a different approach. Indeed, we introduce a new sequence { t k } k of harvest instants, where the time for the ( k + 1)–th capture is determinedby the amount of the biomass harvested at k –th time (which will be denoted by x ( t k )). This leads to a length of closed season determined by(1.2) t k +1 − t k = τ ( x ( t k )) , where τ ( · ) is some function depending on the amount of the biomass captured at t = t k . In consequence, the biomass harvested at time t = t k will determine theinstant of the next capture. The idea is to define a length closure such that a biggercapture leads to a longest closure and by assuming broad and realistic propertieson per capita growth rate and product function, we will find sufficient conditionsensuring a sustainable production, i.e. , the existence of a globally stable periodictrajectory.This approach has been introduced in [9], where it is pointed out that the result-ing model is a new type of IDE equations, namely IDE with impulses dependent oftime (IDE–IDT) and an introductory theory is presented.1.3. Outline.
In section 2, we construct a model of fishery with closed seasonsand pulse captures, which is described by an IDE–IDT. In section 3, we studysome basic properties of the resulting IDE–IDT. The main results concerning thesustainability of the resource are presented in section 4. A numerical example ispresented in section 5. 2.
The model
A classical mathematical model of a fishery with closed seasons has to describethe following bioeconomic processes: i) The growth of the resource ( e.g. , a singlemarine species). ii) The production function. iii) The length of the closures.
Natural rate of growth.
The growth of the biomass in the close season willbe described by the ordinary differential equation:(2.1) x ′ ( t ) = x ( t ) r ( x ( t )) , for any t ∈ ( t k , t k +1 ) , where { t k } denotes an increasing sequence of harvest instants. Growth hypotheses (G)(G1) Density–dependence.
The per capita rate of growth r : [0 , K ] → [0 , + ∞ )is a derivable and strictly decreasing function of the biomass. In addition,we assume r (0) = r > r ( K ) = 0. (G2) Bounded variation. The rate r : [0 , K ] → [0 , + ∞ ) has lowerly boundedderivative, i.e. , there exists ρ >
0, such that − ρ ≤ r ′ ( x ) < Remark . The property of derivability stated in (G1) is a technical assumption.Nevertheless, in [28], some statistical results support the negative correlation be-tween per capita rate of growth and biomass. Notice that, 0 and K are the uniqueequilibria of (2.1) and [0 , K ] is a positively invariant set. Remark . (G2) implies that the function x x r ( x ) satisfies the local Lipschitzcondition when x ∈ [0 , K ]. Hence, the solutions of (2.1) are unique to the right anddepend continuously on initial condition to the right.An example of growth rates satisfying (G) are given by the family:(2.2) r ( x ) = r (cid:16) − h xK i θ (cid:17) β , with θ ≥ β ≥ , which generalizes the logistic Verhulst–Pearl per capita rate (see e.g. , [29]).Another example is given by the function [26]:(2.3) r ( x ) = r K − xK + cx , with c > , which was used, for example, to describe the growth of Daphnia Magna .2.2.
Fishing mortality.
The fishing mortality (see [21, p.102]) is the fraction F ∈ [0 ,
1] of average population taken by fishing. In order to estimate it, weemphasize that there are two possible scenarios: an open season one, where thefishing is allowed all the time, and a restricted process ( closed season ), where thefishing is forbidden.The global fishing process is studied with two time scales: the first one describesthe close season and only considers the growth of biomass summarized by (2.1) withassumptions (G) . The second scale concentrates the fishing mortality by consideringthe captures as pulses defined in a sequence of harvest instants { t k } , obtaining:(2.4) F = x ( t k ) − x ( t + k ) x ( t k ) = H ( x ( t k )) x ( t k ) , where x ( t + k ) is the after k –th capture biomass and H ( · ) describes the capture func-tion in the biomass level at t k . From (2.4), we can see that fishing mortality canbe seen as a measure of catch per unit of biomass (CPUB).For convenience, let us define the impulse action I : [0 , K ] [0 , K ], as follows:(2.5) I ( x ) = x − H ( x ) = (1 − F ) x. In order to relate fishing mortality with the input factors (capital and labor) de-ployed along the fishing process, the continuous modeling literature has introduced the concept of fishing effort , which is a rate describing the number of boats, traps,hooks, technicians, fishermens, etc., per time (see e.g. , [1], [6], [21]).In an impulsive modeling framework, if the punctual fishing effort is denoted by
E >
0, a bounded scalar measure, it is expected to describe (2.4) by a functionalrelation F = φ ( E, x ( t k )), which, combined with (2.1) and (2.5), allows to introducethe complementary evolution law:(2.6) x ( t + ) = I ( x ( t )) = (cid:0) − φ ( E, x ( t )) (cid:1) x ( t ) , with t = t k . We point out that the practical estimation of E and φ ( · , · ) are complicatedmatters in bioeconomic theory and we refer the reader to [1], [6], [30] for details. Harvest hypotheses (H):(H1) Impulsive action . I ( · ) is a derivable and increasing function. (H2) Elasticity . If ∆ x →
0, then:(2.7) ∆
H/H ∆ x/x → H ′ ( x ) xH ( x ) ≥ , for any x ∈ (0 , K ]. This is, a percentage change in the resource biomass de-termines a percentage change bigger than or equal in the captured amount.Notice that, if the yield elasticity respect to the biomass is bigger or equalthan one, then I ( · ) is inelastic or unitary . Remark . The property (H1) combined with (2.6) says that: x ∂φ∂x ( E, x ) ≤ − φ ( E, x ) for any x ∈ [0 , K ] . Remark . The property (H2) says that a fixed punctual fishing effort is moreproductive at higher resource availability. In addition, by using (2.5), we can provethat (H2) is equivalent to I ′ ( x ) xI ( x ) ≤ , and ∂φ∂x ( E, x ) ≥ , for any x ∈ (0 , K ] . Remark . Notice that, the called Cobb–Douglas production function can be in-terpreted by a fishing mortality φ ( E, x ) = qE α x β − with q > α > β > (H) are verified when β ≥
1. An importantcase is the Schaefer assumption [25], corresponding to β = 1 and α = 1, i.e. , theparametrization is linear with respect to the fishing effort and biomass.2.3. Length of the closures.
There exist a third evolution law governing thedynamics, which determine the sequence { t k } of harvest instants. It is the firstorder recurrence that follows:(2.8) ∆ t k = t k +1 − t k = τ ( I ( x ( t k ))) , where τ : [0 , K ∗ ] → [0 , + ∞ ), with K ∗ = I ( K ).As it can be observed in (2.8), the length of the next closure, namely ∆ t k , is afunction τ ( · ) of the stock after the k –th harvest and allows to establish an automaticregulation of the dynamics by closed seasons. Here, we only introduce a theoreticalproposal and we do not deal with the problems of implementation, for instance,those relating to the estimation of data requirred to define the length of the closedseasons. The equation model.
The dynamics determined by the combination of equa-tions (2.1), (2.4) and (2.8) is formalized by the impulsive differential equation:(2.9) x ′ ( t ) = x ( t ) r ( x ( t )) , t = t k ,x ( t + ) = I ( x ( t )) , t = t k , ∆ t k = τ ( I ( x ( t k ))) , k ≥ , where ( t, x ) ∈ [0 , + ∞ ) × [0 , + ∞ ).This type of impulsive differential equation is denoted as Impulsive DifferentialEquations with Impusive Dependent Times (IDE–IDT), which have been intro-duced by C´ordova–Lepe in [9] and its novelty with respect to classic impulsivedifferential equations is that the sequence of impulse instants is determined by theprocess dynamics: the harvested stock I ( x ( t k )) will determine the next harvesttime t k +1 . Indeed, the sequence of harvest times is described by:(2.10) t k +1 = t k + τ ( I ( x ( t k ))) , where the biomass x ( t ) is abruptly reduced to x ( t + ) = x ( t ) − I ( x ( t )) at t = t k .There exists several models of pulse harvesting of a renewable resource (notuniquely restricted to fisheries) described by impulsive equations, e.g. ,: [2], [4], [5],[8] and [32] consider a resource with logistic growth rate, [35] considers a generalizedlogistic growth. In addition, Gompertz models (which, not satisfy (G2) ) have beenstudied in [5], [14]. Nevertheless, these works consider a fixed time between twoharvest processes, which is equivalent to consider τ ( · ) as a constant function.We point out that impulsive models having sequences similar to (2.10) have alsobeen introduced in [20] by Karafyllis in an hybrid control theory setting and arenamed hybrid systems with sampling partition generated by the system .3. The impulsive system (2.9)
Given a first harvest time t ∈ R and a biomass level x ∈ [0 , K ], then theexistence, uniqueness and continuability of the solution of (2.9), with initial condi-tion ( t , x ), can be deduced from [9]. Indeed, we know any solution is a piecewisecontinuous function having first kind discontinuities at the harvest instants t = t k ( k = 0 , , , · · · ). In addition, we point out that different initial conditions willdetermine different sequences of harvest instants.3.1. Basic properties.
In the study of (2.9), it will be necessary to consider theinitial value ordinary associated problem:(3.1) (cid:26) z ′ ( t ) = z ( t ) r ( z ( t )) ,z ( σ ) = v, with ( σ , v ) ∈ R × [0 , K ] . Definition 1.
The unique solution of (3.1) will be denoted by t ϕ ( t ; σ , v ) , forany t ≥ σ , and ϕ ( σ ; σ , v ) = v . Observe that given v ∈ [0 , K ], the function ϕ ( · ; σ , v ) : [ σ , + ∞ ) → [0 , K ] satis-fies:(3.2) ϕ ( t ; σ ,
0) = 0 and ∂ϕ∂v ( t ; σ , v ) ≥ . Let x ( · ) be the solution of (2.9) with initial condition ( t , x ), which determinesthe sequence { ( t k , x ( t k )) } k . Since (2.9) is an ODE on ( t k , t k +1 ], we can deduce that x ( · ) coincides with the unique solution ϕ ( · , t k , I ( x ( t k ))) of (3.1) on ( t k , t k +1 ]. By using Definition 1, it follows that ϕ ( σ + t k ; t k , I ( x ( t k ))) = I ( x ( t k )) exp (cid:16) Z σ + t k t k r (cid:2) ϕ ( s ; t k , I ( x ( t k ))) (cid:3) ds (cid:17) , = I ( x ( t k )) exp (cid:16) Z σ r (cid:2) ϕ ( s + t k ; t k , I ( x ( t k ))) (cid:3) ds (cid:17) . Finally, uniqueness of solutions implies ϕ ( s + t k ; t k , I ( x ( t k ))) = ϕ ( s ; 0 , I ( x ( t k ))),which leads to:(3.3) ϕ ( σ + t k ; t k , I ( x ( t k ))) = I ( x ( t k )) exp (cid:16) Z σ r (cid:2) ϕ ( s ; 0 , I ( x ( t k ))) (cid:3) ds (cid:17) , for any σ ∈ [0 , τ ( I ( x ( t k ))].By using (2.10), it follows that at σ = τ ( I ( x ( t k ))) ( i.e. , at t = t k +1 ), the solutionsof (2.9) satisfy the one dimensional map:(3.4) x ( t k +1 ) = f ( x ( t k )) , where the function f : [0 , K ] → [0 , K ] is described as follows:(3.5) f ( x ) = F ( I ( x )) , with F ( y ) = y exp (cid:16) Z τ ( y )0 r [ ϕ ( s ; 0 , y )] ds (cid:17) . Notice that Eq.(2.6) implies f (0) = 0. This fact will have important conse-quences when studying the asymptotic behavior of (2.9).3.2. Special solutions of (2.9) and bio-economic interpretation.
Let us in-troduce the straightforward result:
Lemma 1.
Any positive fixed point u ∗ ∈ (0 , K ] of the map (3.4) defines a piecewise–continuous τ ( I ( u ∗ )) –periodic solution t u ∗ ( t ) of (2.9) , namely, the u ∗ –associatedsolution. In addition, the fixed point u ∗ = 0 of the map (3.4) defines a constantnull solution of (2.9) . The existence of a τ ( I ( u ∗ ))–periodic solution can be interpreted as a fisherystrategy with harvest instants uniformly distributed in time. There exists differentstability definitions for these solutions (see e.g. , [9] and [20]). In this context, wewill follow the ideas stated in [10]: Definition 2.
The u ∗ –associated solution of (3.4) is locally asymptotically stable ifthere exists δ > such that for any solution t x ( t ) of (2.9) with initial condition x (0) satisfying | x (0) − u ∗ | < δ , it follows that: lim k → + ∞ | x ( t k ) − u ∗ | = 0 , where { t k } is the corresponding sequence of harvest instants associated to x (0) . Definition 3.
The u ∗ –associated solution of (3.4) is globally asymptotically stableif for any solution t x ( t ) of (2.9) with initial condition x (0) > , it follows that: lim k → + ∞ | x ( t k ) − u ∗ | = 0 . Observe that the asymptotic stability of a τ ( I ( u ∗ ))–periodic solution implies theecological sustainability of the fishery. On the other hand, the asymptotic stabilityof the null solution implies the future resource extinction. Sustainability conditions
General result.Theorem 1.
Let us assume that (G) , (H) and the closed season hypotheses: (C1) Growth type. The function τ : [0 , K ∗ ] → [0 , + ∞ ) is derivable and de-creasing, such that τ ( K ∗ ) = 0 . (C2) Initial condition . The initial value τ = τ (0) satisfies (4.1) τ ≤ ρK + r ln (cid:16) − φ ( E, (cid:16) r ρK (cid:17)(cid:17) . (C3) Main condition . The following inequality: (4.2) | τ ( z ) − τ ( z ) | < r (cid:26) ln (cid:18) z z (cid:19) − m ( z − z ) (cid:27) , is verified for < z < z < I ( K ) = K ∗ and m = ρ [ e ατ − /α , with α = ρK + r .are satisfied.Then: i) If e τ r (1 − φ ( E, < , then the resource–free solution of (2.9) is globallyasymptotically stable ( extinction case ) . ii) If e τ r (1 − φ ( E, > , then there exists a unique initial condition x ∗ = f ( x ∗ ) ∈ (0 , K ) –with f ( · ) given by (3.4) – defining a τ ( I ( x ∗ )) –periodic glob-ally asymptotically stable trajectory ( sustainable case ) .Remark . Notice that (C3) gives us a range of graphic possibilities for the choiceof function τ ( · ). Moreover, (4.1) implies that the right side of (4.2) is non negativefor any z ∈ [0 , K ∗ ]. Remark . If (C2) is verified, we can see that e τ r (1 − φ ( E, > r | ln(1 − φ ( E, | < τ ≤ ρK + r ln (cid:16) − φ ( E, (cid:16) r ρK (cid:17)(cid:17) . In consequence, the left inequality says that there exists a trade–off between thefishing effort E and the maximal lenght of a closure τ ensuring the fishery sus-taintability. In addition, observe that the inequality stated above has sense onlywhen φ ( E, ∈ [0 , φ ∗ ) ⊂ [0 , Proof.
The asymptotic behavior of (2.9) is determined by the map (3.4).Now, we will verify that f : [0 , K ] → R satisfies the following properties.a) The map f is derivable and f (0) = 0,b) The map f is increasing and f ( K ) < K ,c) For any x ∈ (0 , K ] it follows that:0 ≤ xf ′ ( x ) f ( x ) < . Indeed, a) is straightforward consequence from (2.6) combined with f ′ ( x ) = F ′ ( I ( x )) I ′ ( x ), where F ′ ( y ) is defined by:exp (cid:16) Z τ ( y )0 r [ ϕ ( s ; 0 , y )] ds (cid:17)n y (cid:16) r [ ϕ ( τ ( y ); 0 , y )] τ ′ ( y ) + Z τ ( y )0 ∂r∂y [ ϕ ( s ; 0 , y )] ds (cid:17)o . Let us verify that f ′ (0) = e τ r I ′ (0) is consequence from (G1) , (H1) , (C2) and(3.2). Now, by using (H1) , we observe that b) follows if F ′ ( y ) > y ∈ [0 , K ].When dropping the exponential factor of F ′ ( · ), we only have to prove that(4.3) 1 + y (cid:16) r [ ϕ ( τ ( y ); 0 , y )] τ ′ ( y ) + Z τ ( y )0 ∂r∂y [ ϕ ( s ; 0 , y )] ds (cid:17) > y ∈ [0 , K ].By hypotheses (G) and (C1) , combined with ∂ϕ∂y ( s ; 0 , y ) > s ∈ [0 , τ ( y )],we can observe that inequality (4.3) can be deduced from:(4.4) 1 > y (cid:16) r | τ ′ ( y ) | + ρ Z τ ( y )0 ∂ϕ∂y ( s ; 0 , y ) ds (cid:17) . By integral representation of ϕ ( s ; 0 , y + h ) and ϕ ( s ; 0 , y ), with s ∈ [0 , τ ( y )] and h > | ϕ ( s ; 0 , y + h ) − ϕ ( s ; 0 , y ) | ≤ | h | e ( ρ K + r ) s , for any s ∈ [0 , τ ( y )]. Indeed, ∂ϕ∂y ( s ; 0 , y ) ≤ e ( ρK + r ) s , with s ∈ [0 , τ ( y )]. Therefore,we can reduce our proof to demand the condition that follows:(4.5) y (cid:16) ρα h e ατ ( y ) − i − r τ ′ ( y ) (cid:17) < , where α = ρK + r . We point out that (C3) is equivalent to (4.5). Indeed, when replacing z and z by y and y + h ( h >
0) respectively, (4.5) is obtained by letting h →
0. Inversely,since τ ( y ) < τ = τ (0), the inequality (4.2) is obtained by integrating (4.5) on[0 , K ∗ ], with K ∗ = I ( K ).To prove that the right side of (4.2) is greater than zero, we observe thatinf n ln( K ∗ ) − ln( z ) K ∗ − z : z ∈ (0 , K ∗ ] o = 1 K ∗ , and 1 /K ∗ > m is equivalent to (C2) . Finally, observe that (H) and (C1) imply f ( K ) = I ( K ) < K and property b) follows.The property c) is equivalent to x ( f ( x ) /x ) ′ = F ′ ( I ( x )) I ′ ( x ) x − F ( I ( x )) < x ∈ (0 , K ]. This is verified if and only if for any x ∈ (0 , K ], it follows that:(4.6) x I ′ ( x ) [1 + I ( x ) W ′ ( I ( x ))] exp( W ( I ( x ))) ≤ I ( x ) exp( W ( I ( x )))where W ( I ( x )) is defined by W ( I ( x )) = Z [0 ,τ ( I ( x ))] r [ ϕ ( s ; 0 , I ( x ))] ds, for x ∈ [0 , K ] . Let us recall that W ′ ( y ) = r [ ϕ ( τ ( y ); 0 , y )] τ ′ ( y ) + Z τ ( y )0 ∂r∂y [ ϕ ( s ; 0 , y )] ds < , for y ∈ (0 , K ∗ ] . By using this inequality, combined with (4.3) and Remark 4, we can deduce that:(4.7) I ′ ( x ) xI ( x ) ≤ < − I ( x ) | W ′ ( I ( x )) | , with x ∈ (0 , K ] . By applying Lemma 2 (see Appendix), there are two posibilities for system (3.4)according the sign of 1 − f ′ (0) = 1 − e τ r I ′ (0) . By Eq.(2.6), we can express the threshold condition for case (a) f ′ (0) < e τ r (1 − φ ( E, < f ′ (0) > e τ r (1 − φ ( E, >
1. So that,the result follows. (cid:3)
Application to logistic growth.
Notice that, in some cases, the one–di-mensional map (3.4) associated to the system (2.9) can be defined explicitly andthe previous result improved. Indeed, let us consider a marine species with logisticgrowth, whose exploitation is described by:(4.8) x ′ ( t ) = rx ( t ) (cid:16) − x ( t ) K (cid:17) , t = t k ,x ( t + ) = I ( x ( t )) , t = t k , ∆ t k = τ ( I ( x ( t k ))) , k ≥ . Corollary 1.
Let us assume that the impulse action I ( · ) satisfies (H) and theclose season satisfies (C1) and (C3’) Closure condition . The following inequality: | τ ( z ) − τ ( z ) | < r (cid:26) ln (cid:18) z z (cid:19) + ln (cid:16) K − z K − z (cid:17)(cid:27) , is verified for < z < z < I ( K ) = K ∗ .Then: i) If e rτ (1 − φ ( E, < , then the resource–free solution of (4.8) is globallyasymptotically stable ( extinction case ) . ii) If e rτ (1 − φ ( E, > , then there exists a unique initial condition x ∗ = f ( x ∗ ) ∈ (0 , K ) defining a τ ( I ( x ∗ )) –periodic globally asymptotically stabletrajectory ( sustainable case ) .Proof. A simple computation shows that (3.4) is equivalent to the one–dimensionalmap:(4.9) x ( t k +1 ) = f ( x ( t k )) = KI ( x ( t k )) I ( x ( t k )) + [ K − I ( x ( t k ))] e − rτ ( I ( x ( t k ))) . We will verify that the map (4.9) satisfies the assumptions of Lemma 2 (seeAppendix). Firstly, notice that f ( · ) can be described as follows: f ( x ) = F ( I ( x )) , with F ( u ) = Kuu + [ K − u ] e − rτ ( u ) , and by using (H) , it follows that f ( · ) is derivable and f (0) = 0.Secondly, observe that f ′ ( x ) = F ′ ( I ( x )) I ′ ( x ). As in the proof of Theorem 1,it follows that F ′ ( u ) > ≤ u ≤ I ( K ) = K ∗ ) if and only if (C3’) isverified. By using this fact, combined with (H1) , it follows that f ( · ) is increasingand f ( K ) = F ( I ( K )) < K .Finally, from (H2) combined with Remark 4, it follows that0 ≤ xf ′ ( x ) f ( x ) = xI ′ ( x ) F ′ ( I ( x )) F ( I ( x )) = xI ′ ( x ) I ( x ) F ′ ( I ( x )) I ( x ) F ( I ( x )) ≤ F ′ ( I ( x )) I ( x ) F ( I ( x )) . By using (C1) , it is not difficult to show that uF ′ ( u ) < F ( u ) for any u ∈ (0 , I ( K )]. This fact, combined with the last above inequality implies that 0 ≤ xf ′ ( x ) /f ( x ) < x ∈ (0 , K ]. Now, as f ′ (0) = e rτ (1 − φ ( E, (cid:3) Remark . i) The assumption (C3’) is equivalent to the differential inequality K > − ru ( K − u ) τ ′ ( u ) for any 0 ≤ u ≤ I ( K ), which furnishes a way to designadmissible functions τ ( · ) describing the lengh of open seasons.ii) In addition, it is easy to verify that in this case there are no explicit re-strictions for τ as stated in (C2) (see also Remark 6).5. Example
Let us consider a fishery with: biomass growth described by the logistic equation,fishing mortality satisfying Schaefer assumption, i.e. , φ ( E, x ) = qE and closureshaving lengths determined by the linear function τ : [0 , K ∗ ] → [0 , + ∞ ):(5.1) τ ( z ) = a ( K ∗ − z ) , with a > . By using remarks 1 and 5, it follows that hypotheses (G) and (H) are satisfied.In addition, observe that assumption (C1) is satisfied since τ ( · ) is strictly decreasingand τ ( K ∗ ) = 0. Moreover, notice that: − ru ( K − u ) τ ′ ( u ) = aru ( K − u ) ≤ arK / , and by using statement i) from Remark 8, it follows that (C3’) is satisfied if a < / ( rK ).On the other hand, as K ∗ = (1 − qE ) K and τ = τ (0) = a (1 − qE ) K , we obtainthe threshold:(5.2) E ( qE ) = e rτ (1 − φ ( E, e arK (1 − qE ) (1 − qE ) , and by Corollary 1, it follows that the resource–free solution of (4.8) is globallyasymptotically stable if E ( qE ) <
1. Similarly, there exists a globally asymptoticallystable periodic solution if E ( qE ) > , E ∗ )( E ( qE ∗ ) = 1) ensuring sustainability, let us represent the slope of (5.1) as follows: a = 4 rK η, with 0 < η < . Notice that, to find E ∗ is equivalent to find the unique fixed point w ∗ = 1 − qE ∗ of the map w e − ηw . In this case, E ∗ is dependent of the parameter η ∈ (0 , τ .It is straightforward to verify that the function η E ∗ is increasing and concave.This implies that lower values of maximal closure length τ leads to narrow rangesof sustainable punctual effort (0 , E ∗ ).We illustrate this previous results by using the numerical methods developed byDel-Valle [12]. The following parameters are employed:(5.3) r = 0 .
05 [time − ] , K = 1000 [tons] , q = 1 and η = 0 . . These values determine w ∗ ∼ . E ∗ ∼ . E . . . . x ( t ) Biomass vs Time E=0.1000E=0.4000E=0.4329E=0.5000
Figure 3.
Numerical solution of (4.8) with Schaefer assumption,closure length defined by (5.1) and parameters (5.3). The punctualfishing efforts ensuring sustainability satisfy E ∈ (0 , . x (0) = 300 tons and four different punctual fishing efforts. As stated before,we can see an extinction scenario for any punctual effort bigger than E ∗ (this isthe case for E = 0 . E .6. Discussion
The mains results (Theorem 1 and Corollary 1) propose sufficient conditionsensuring the ecological sustainability of a simple fishery model with impulsive cap-tures, which can be seen as the trade–off between the fishing effort E and themaximal closure length τ . The novelty (compared with harvest instants uniformlydistributed in time) is to allow a variable length of closures, which has social andeconomic consequences in a short term.The proof of Theorem 1 is carried out by constructing a one–dimensional map(3.4), whose asymptotic behavior inherits crucial features of the IDE–IDT equa-tion. In this context, this approach could be extended in several ways. First, noticethat assumptions (C2) and (C3) imply that the map (3.4) is strictly increasing.Whe think that it is possible to consider more general maps and obtain less restric-tive conditions ensuring convergence towards a fixed point. This remains a futureproblem and its main difficulty is the hybrid nature of IDE–IDT equations.Provided that the fishery is sustainable, an important open problem is to findfirst order conditions on the punctual fishing effort E that maximizes: φ ( E, x ∗ ( E )) x ∗ ( E ) τ ([1 − φ ( E, x ∗ ( E ))] x ∗ ( E )) , the sustainable production per unit time. In other words, the task is the MaximunSustainable Yield (MSY) problem.Another extension of this work would be to consider the logistic case (4.8),replacing r > r : R → R + . This problem is interesting by a bioeconomic point of view since periodic and Bohr almost periodic functionsprovide a good tool in order to modeling birth rates with seasonal behavior.On the other hand, to consider the logistic model (4.8) with almost periodicperturbations has mathematical interest in itself. Indeed, if r ( · ) is a positive Bohralmost periodic function and τ ( · ) is a constant function, it can be proved thatthere exist a unique almost periodic solution (in the sense of Samolienko and Per-estyuk [24]), which is globally asymptotically stable. Nevertheless, there are neitherequivalent results nor a study of asymptotic properties when τ ( · ) is not constant. Acareful development of the qualitative theory for IDE–IDT seems essential to copewith this kind of problems. Appendix
The following result plays a key role in the proof of Theorem 1:
Lemma 2.
Let us consider the one dimensional map: (6.1) x n +1 = f ( x n ) , x ∈ [0 , K ] , where the function f : [0 , K ] [0 , f ( K )] ⊂ [0 , K ) satisfies the following properties: a) f is derivable and f (0) = 0 , b) f is increasing and f ( K ) < K , c) For any x ∈ (0 , K ] it follows that: ≤ xf ′ ( x ) f ( x ) < . i) If f ′ (0) < then it follows that lim n → + ∞ x n = 0 . ii) If f ′ (0) > then there exists a unique fixed point x ∗ ∈ (0 , K ) a it followsthat lim n → + ∞ x n = x ∗ for any x ∈ (0 , K ] .Proof. Case f ′ (0) <
1: we can verify that there exists δ > f ( x ) < x forany x ∈ (0 , δ ). Let us denote by A , the set of positive fixed points of f . If A = ∅ ,by using continuity of f ( · ) it follows that f ( x ) < x for any x ∈ (0 , K ].If A = ∅ , let us define ξ = inf { A } . It is straightforward to verify that:(6.2) ξ > f ( x ) < x for any x ∈ (0 , ξ ) . Now, by using c) we can deduce that 0 ≤ f ′ ( ξ ) <
1, which implies the existenceof δ > f ( x ) ≥ x for any x ∈ ( ξ − δ, ξ ), obtaining a contradiction with(6.2).In consequence 0 is the unique fixed point in [0 , K ] and f ( x ) < x implies that { x n } is strictly descreasing and lowerly bounded. Finally, the convergence towards x = 0 follows from uniqueness of the fixed point. Case f ′ (0) >
1: We can verify that there exists δ > f ( x ) > x for any x ∈ (0 , δ ). On the other hand, f ( K ) < K combined with continuity of f imply theexistence of a fixed point x ∗ ∈ (0 , K ) minimal with this property. Hence, it followsthat f ( x ) > x for any x ∈ (0 , x ∗ ) and (c) implies that 0 ≤ f ′ ( x ∗ ) < x ∗ , let us define g ( x ) = f ( x + x ∗ ) − x ∗ andobserve that g : [0 , K − x ∗ ] [0 , f ( K ) − x ∗ ] ⊂ [0 , K − x ∗ ) and g ′ (0) <
1. Hence,the non–existence of fixed points on ( x ∗ , K ) and the property f ( x ) < x for any x ∈ ( x ∗ , K ] follows as in the previous case. If x ∈ ( x ∗ , K ] it follows that { x n } is strictly descreasing and lowerly bounded by x ∗ . On the other hand, if x ∈ (0 , x ∗ ) it follows that { x n } is strictly increasing andupperly bounded by x ∗ . The convergence towards x ∗ follows from the uniquenessof the fixed point in (0 , K ). (cid:3) Example : The function f : [0 , K ] → R defined by f ( x ) = λx/ ( a + x ) with a > λ < a satisfies straightforwardly the properties a) and b) by choosing K > a .Finally, observe that:0 ≤ xf ′ ( x ) f ( x ) = 1 a + x < , for any x ≥ , and c) is verified.Hence, if λ < a ( i.e. f ′ (0) < { x n } n definedrecursively by (6.1) with x ∈ (0 , K ] is convergent to 0. Otherwise, if λ ∈ ( a, a )( i.e , f ′ (0) > x ∗ = a − λ . Ackowledgements
We would like to express our thanks to professor Luis Cubillos(Universidad de Concepci´on – Chile) for the data contained in Figure 1. The firstauthor acknowledges the support of Direcci´on de Investigaci´on UMCE.
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