Optimality and Sustainability of Delayed Impulsive Harvesting

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Jennifer Lawson and Elena Braverman

Dept. of Math & Stats, University of Calgary,

2500 University Dr. NW, Calgary, AB, Canada T2N1N4

Abstract

We consider a logistic diﬀerential equation subject to impulsive delayed harvesting, where the

deduction information is a function of the population size at the time of one of the previous

impulses. A close connection to the dynamics of high-order diﬀerence equations is used to conclude

that while the inclusion of a delay in the impulsive condition does not impact the optimality of

the yield, sustainability may be highly aﬀected and is generally delay-dependent. Maximal and

other types of yields are explored, and sharp stability tests are obtained for the model, as well as

explicit suﬃcient conditions. It is also shown that persistence of the solution is not guaranteed

for all positive initial conditions, and extinction in ﬁnite time is possible, as is illustrated in the

simulations.

Keywords: Optimal harvesting; logistic equation; impulsive system; impulsive delayed har-

vesting; population dynamics

AMS (MOS) subject classiﬁcation: 92D25, 34A37

1. Introduction

Optimal sustainable resource management is one of the most important modern problems, with

the dynamics of harvesting models being an essential facet [1]. Harvesting can be represented in

models either continuously or as only occurring during short-time periods. Continuous harvesting

can be described as a continuous deduction term appearing in an ordinary Diﬀerential Equation

(DE) determining population dynamics. It assumes that harvesting occurs without any interrup-

tions, whereas impulsive harvesting corresponds to part of the stock being removed at speciﬁc

moments in time, with the duration of the harvesting event being negligible compared to the pro-

cess time. Even though continuous harvesting may be preferable from the point of view of both

maximizing harvest and sustainability [2, 3], it is not always realistic or easily applicable. This is

why investigation of impulsive harvesting models is important.

Impulsive DEs incorporate two parts, the DE that describes the behaviour of the system during

times of continuous dynamics, and the conditions that govern the instantaneous change in the

system at the impulsive moments. Usually, this instantaneous change is a result of some external

eﬀect on the system, which has a duration that is negligible compared to the overall time scale of the

process. Impulsive DEs have many practical applications such as pest control [4], pulse vaccination

strategies [5], and optimal harvesting in ﬁsheries [6]. For more on the theory of impulsive DEs see

the monograph [7].

It is well known that including delays within a DE model of population dynamics can lead to

major changes to its behaviour, such as causing instability, oscillations, and extinction, which are

Preprint submitted to Communications in Nonlinear Science and Numerical Simulation Final version

arXiv:2210.05878v1 [math.DS] 12 Oct 2022

not observed in a corresponding ordinary DE model [8]. The famous Hutchinson equation is one

such example. It can be compared to the logistic equation where the carrying capacity is always a

globally attractive equilibrium for all non-trivial positive solutions. The inclusion of a delay in the

Hutchinson equation can cause the carrying capacity equilibrium to become unstable for certain

values of a delay. Other examples speciﬁc to harvesting models can be seen in [9, 10], where delay

is incorporated into the continuous harvesting terms.

In an impulsive system, whenever control is involved, it is natural to assume that the informa-

tion available at the control implementation point is not up-to-date, leading to delayed impulsive

conditions. The incorporation of delays in impulses goes back to the 1990s [11] with the further

development of non-instantaneous impulse theory, some of which is summarized in the mono-

graphs [12, 13], and with recent progress reported in [14, 15, 16]. Delayed impulses in the context

of harvesting were explored in [17]. To the best of our knowledge, there still has been no investiga-

tion of the optimal harvesting policies and their sustainability for systems with delayed impulsive

harvesting in the literature, and we aim to ﬁll this gap.

For logistic and other simple population models, such as Gompertz, incorporating stochastic

ﬂuctuations or random diﬀerential equations created an additional challenge but reﬂected unpre-

dictable changes in the environment, see [18, 19, 20] and the references therein. Impulsive harvesting

of a stochastic Gompertz model was a focus of [19]. Though logistic-type equations are considered

in most studies on optimal control, the use of other growth rates can modify the preferred policies

[21]. The close connection of continuous models to diﬀerence equations has led to extensive study

of discrete population models [22] including harvesting [23]. The fact that species do not naturally

exist in isolation but coexist, compete or serve as prey for others, led to extensive literature on

harvesting of a single or multiple populations in a food chain, and on optimal yields for exploited

species [24, 25, 26, 27]. Incorporating harvesting in systems of diﬀerential or diﬀerence equations

includes the case of structured populations where selective harvesting is allowed [28, 29]. This

approach is quite useful in policy-making, for example, when only ﬁsh types within a certain size

range are eligible to take-home vs catch-and-release policies.

We consider the logistic equation with constant catch-per-unit eﬀort impulsive harvesting that

is dependent upon delayed data. This assumes that the information used to determine hunting or

ﬁshery quotas is based on data for the population size or structure which was collected during one

of the previous harvesting events.

The main object of the present paper is the delayed impulsive harvesting model, given for a

ﬁxed k∈Nas











dt =rN(t)1−N(t)

Kc, t 6=nT, n ∈N

N(nT +) = max{N(nT )−EN((n−k)T),0}, t =nT, n ∈N

N(0+) = N+

0, N(0) = N0, ..., N (−(k−1)T) = N−(k−1)

(1)

with prescribed initial conditions

N+

0, Ni∈(0,∞), i =−(k−1), ..., 0.

In this model, the left-continuous N(t) represents a size or a biomass of the population as a function

of time, r > 0 is the intrinsic growth rate, Kc>0 is a carrying capacity of the environment, T > 0

is the time between two consecutive harvesting events, Eis a harvesting eﬀort and is assumed to

be E∈(0,1) to avoid immediate extinction. We assume both that restocking does not occur, and

that the function N(t) is left continuous. We denote the size of the population after harvesting as

lim

h→0+N(nT +h) = N(nT +), whereas the size of the population N(t) before harvesting is N(nT ), for

the left-continuous function N. Note that we do not require continuity at zero i.e. N(0+) = N(0),

nor do we assume anything about the size of the population between t=−iT and t=−(i−1)T.

This makes room for the possibility that the ﬁrst harvesting event occurs at t= 0, as might be

likely in a real-world setting, but does not require that this happens. We assume k∈N, though

references to the non-delay model with k= 0 will also be given.

The motivation of incorporating delay in the impulsive harvesting is two-fold:

1. Impulsive harvesting in general allows us to describe short duration harvesting events, the

eﬀect of which makes it impossible for population growth to remain continuously at optimal

levels. It is a well developed topic, with multiple publications appearing in the literature

[2, 6, 17, 3, 30], but not for delayed impulsive conditions. Thus, delayed impulsive models

are of theoretical interest. While the consideration of delayed impulsive systems is not new

[11], recent developments in the theory of such dynamical systems [14, 15, 16] have made its

practical study feasible, as there are theoretical tools to investigate it.

2. Harvesting policies are dependent upon population data, and often the data which is used is

not up-to-date, leading to a delay in the harvesting term of models. It is therefore important

to be able to assess the impact of the delay, so that managers may determine what addi-

tional modiﬁcations must be made to the harvesting decisions. While non-delay harvesting

models depict gradual declines to extinction, they mostly failed to explain a frequently ob-

served phenomenon of instantaneous species disappearance due to over-exploitation. Unlike

traditional continuous or non-delayed impulsive harvesting, delayed impulsive harvesting can

describe immediate, not long-term collapse of harvested population. Truncated models where

the population or commodity size is chosen as a maximum of the computed value or zero,

are quite common in mathematical economics and discrete dynamical systems. We found it

natural to extend this method to population dynamics in order to describe possible extinction

in ﬁnite time.

Let us also dwell on the choice of the model. Our purpose was to explore and emphasize the

eﬀect of the harvesting delay in the impulsive form, thus we consider the simplest autonomous

logistic equations. The results will outline the intrinsic eﬀect of the delay in short-term harvesting.

Our goal is ﬁrst, to consider the sustainability of (1) under harvesting, which corresponds to

the local asymptotic stability of a positive solution which will be described later, and second, to

explore the sustainable yield (SY) and the maximum sustainable yield (MSY) of (1). The paper

is structured to follow this purpose. After presenting relevant deﬁnitions and auxiliary results in

Section 2, we explore stability of (1) in Section 3. All the issues connected to SY and MSY, and

relevant solutions of (1), are postponed to Section 4. We show that, while optimality is unaﬀected

by the magnitude of delay, sustainability of the optimal solution is delay-dependent for k≥2. The

analysis of the impact of the delay on local asymptotic stability of the positive solution is based on

the results obtained in Section 3. Finally, Section 5 includes examples, numerical simulations, as

well as discussion of the results of the paper and possible future directions.

2. Preliminaries and Auxiliary Results

A solution N∗(t) of impulsive system (1) is said to be stable if for any ε > 0 there exists a

δ=δ(ε)>0 such that the inequalities |Nj−N∗(jT )|< δ,j=−k+ 1,...,0, |N+

0−N∗(0+)|< δ

imply |N(t)−N∗(t)|< ε for all t > 0. A solution of (1) is (locally) asymptotically stable if it

is stable and there exists η > 0 such that lim

t→∞ |N(t)−N∗(t)|= 0 for any |Nj−N∗(jT )|< η,

j=−k+ 1,...,0, |N+

0−N∗(0+)|< η. A solution of (1) is globally asymptotically stable if it is

stable and lim

t→∞ |N(t)−N∗(t)|= 0 for any N+

0>0, N0, N−1..., N−k+1 ≥0.

Ak-th order diﬀerence equation has the form

xn+1 =f(xn, xn−1, ..., xn−k), n ∈N, n ≥k(2)

with the initial conditions x0, ..., xk.

A solution xn≡x∗of (2) is stable if for all ε > 0, there exists a δ > 0 such that max{|x0−

x∗|, ..., |xk−x∗|} < δ implies |xn−x∗|< ε for any n≥k. A solution x∗of (2) is (locally)

asymptotically stable if it is stable and there exists η > 0 such that max{|x0−x∗|, ..., |xk−x∗|} < η

implies lim

n→∞ |xn−x∗|= 0. A solution x∗of (2) is globally asymptotically stable if it is stable and

lim

n→∞ |xn−x∗|= 0 for any x0, ..., xk.

If harvesting is restricted to only the surplus production of a population, then theoretically,

harvesting should be able to continue indeﬁnitely without drastically altering the stock levels. The

idea behind the Maximum Yield (MY) is that it corresponds to an optimal solution of (1) such that

the yield will not be exceeded by any other solution, and that the optimal solution is T-periodic

leading to a constant yield over a time period. A Maximum Sustainable Yield (MSY) is a MY

where the optimal solution of (1) is (at least locally) asymptotically stable.

In [30], the authors considered an MSY for (1) with k= 0. The results are summarized in the

following.

Lemma 2.1. [30] Consider (1) for k= 0











dt =rN(t)1−N(t)

Kc, t 6=nT, n ∈N

N(nT +) = max{(1 −E)N(nT ),0}, t =nT, n ∈N

N(0) = N0.

(3)

Then the optimal harvesting eﬀort is

Eopt = 1 −e−rT/2,(4)

and the MSY is given by

MSY =Kc(erT /2−1)

T(erT /2+ 1) (5)

The optimal positive periodic solution N∗(t)of (3) corresponding to the MSY and Eopt satisﬁes

N∗(nT +) = Kc

erT /2+ 1 (6)

and is globally asymptotically stable.

In [30] analysis of a non-delayed impulsive model (3) is reduced to a nonlinear diﬀerence equa-

tion of the ﬁrst order. We also intensively exploit connection between diﬀerence and impulsive

equations. When a delay is incorporated in impulsive condition, the diﬀerence equation becomes

higher order. We recall that for diﬀerence equations, the roots of the characteristic equation of an

associated linearized model should lie inside the unit circle for local asymptotic stability, in con-

trast to diﬀerential equations where the real parts of the roots have to be negative. Some auxiliary

results regarding diﬀerence equations are listed below.

Lemma 2.2 ([31]).The roots of the characteristic equation

p(λ) = λ2−p0λ+p1, p0>0, p1>0

lie inside the unit circle if and only if p0−1< p1<1.

The result of [32, Theorem 1.1.1, Part f, P. 7] describes conditions when a root of a quadratic

equation lies on the boundary of the unit circle.

Lemma 2.3 ([32]).A necessary and suﬃcient condition for a root of the characteristic equation

λ2−p0λ+p1= 0

with p0, p1∈Rto have a root satisfying |λ|= 1 is that either

|p0|=|1 + p1|

p1= 1 and |p0| ≤ 2.

Lemma 2.4 is cited from [31, Theorem 5.10, P. 253].

Lemma 2.4 ([31]).If

i=0

|pi|<1then the zero solution of the diﬀerence equation

xn+k+1 +p0xn+k+p1xn+k−1+... +pkxn= 0

is asymptotically stable.

The following result can be found in [31, Theorem 5.3, P. 249].

Lemma 2.5 ([31]).Let p0>0,pk∈Rbe arbitrary, and k∈N. The zero solution of the equation

xn+1 −p0xn+pkxn−k= 0 (7)

is asymptotically stable if and only if |p0|<(k+ 1)/k and

(i) |p0| − 1< pk<(p2

0+ 1 −2|p0|cos(θ∗))1/2if kis odd

(ii) |pk−p0|<1and |pk|<(p2

0+ 1 −2|p0|cos(θ∗))1/2if kis even,

where θ∗is the solution of the equation

sin(kθ)

sin((k+ 1)θ)=1

|p0|, θ ∈0,π

k+ 1.(8)

However, we do not need the general form of Lemma 2.5, since in our model 0 < pk< p0. Then

the left inequality in both (i) and (ii) becomes p0< pk+ 1, the right inequalities coincide.

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OptimalityandSustainabilityofDelayedImpulsiveHarvestingJenniferLawsonandElenaBravermanDept.ofMath&Stats,UniversityofCalgary,2500UniversityDr.NW,Calgary,AB,CanadaT2N1N4AbstractWeconsideralogisticdierentialequationsubjecttoimpulsivedelayedharvesting,wherethedeductioninformationisafunctionofthepopulat...

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