Delay Differential Logistic Equations with Harvesting

N(t) = r(t)N(t) a ~ bkN(hk(t)) cl(t)N(gl(t)), k ~ l l = l N(t)=~(t) , t O , is considered. The existence and the bounds of positive solutions are studied. Sufficient conditions for the extinction of the solution are presented. © 2004 Elsevier Ltd. All rights reserved. K e y w o r d s D e l a y logistic equations, Linear harvesting, Positive solutions, Extinction of the population, Solution bounds. 1. I N T R O D U C T I O N The simplest model of the population growth is described by t'he equation d N dt = r N ( t ) . *Partially supported by Israeli Ministry of Absorption. tAuthor to whom all correspondence should be addressed. Partially supported by the NSERC Research Grant and the AIF Research Grant. :~Partially supported by MRF (Malaspina University-College Grant). 0895-7177/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm. 2004.06.005 Typeset by A.A, q S ~ 1244 L. BEREZANSKY et al. In more sophisticated models of population dynamics, the growth rate is assumed to be a function of the population N. If the growth rate is denoted by r(N) , then the differential equation describing population growth takes the form d N dt = r (N( t ) , t )N( t ) . It is legitimate to regard r(N( t ) , t) as formally describing a self-regulatory mechanism. Of course, most populations do not live a life unmolested by outside influences. We will study the dynamics of a population affected by harvesting. The following general differential equation d N dt = r(N( t ) , t )N( t ) g ( N ( t ) , t) (1) will be considered, where E(N , t) is a harvesting strategy for the population. Function E represents the rate at which individuals are harvested. In 1959, Holling [1] identified three basic types of functional "predator" responses: • Type I (linear): E ( N , t ) = a N +~, • Type II (cyrtoid): E(N , t) = a N / ( N + ~), • Type III (sigmoid): E(N , t) = aN2/(V 2 + fiN 2) ( a, ~, and V are positive functions of t). In this paper, we will focus on the response of Type I. If r (N) = N(a bN), where a > 0, b > 0 are constants, then we have a logistic-like equation with the harvesting strategy d N dt = N ( a bN) E ( N , t). (2) Below, some applied models of population dynamics are described. EXAMPLE 1.1. METAPOPULATION MODEL. One of the first metapopulation models was developed in 1969 by Levins [2]. It considered immigration of organisms (e.g., birds) from a continent to islands in the ocean. The proportion of islands colonized by a species, N, is given by a natural logistic growth term offset by losses linearly proportional to the birds population (Type I) dNdt = c( t )N ( 1 N ) e ( t ) N ' where c is a colonization rate, e is an extinction rate. EXAMPLE 1.2. LOBSTER FISHERY. For fishery management and many other harvesting situations, it is unreasonable to assume that the harvesting rate is constant (i.e., independent of the population). Thus, for modeling the marine fishery, we will assume that the harvesting rate H is proportional to the population N (Type 1). The differential equation [3-5] is then d N dt r N ( K N) Q C N , where Q is a biological characteristic of the population which is not subject to human control, while C is under the control of the lobster fishery. Note that a loss rate due to harvesting, in general, depends both on the fishing effort and on the fish population density. The same is true for whaling industries and commercial forestries. The last few decades have seen an expanding interest in retarded models of population dynamics involving logistic differential equations with delay. There have been many attempts [6-10] to find reasonable mathematical models with time lags to describe certain complex biological systems. For example, the delay logistic differential equation in population ecology d N r ( t )N( t ) ( N(h(t)) , h(t) < Delay Differential Logistic Equations 1245 is known as Hutchinson's equation [11], where r and K are positive constants and h( t ) = t T with a positive constant T. The delays are used in immunology to represent the time needed for immune cells to divide, or become destined to die. In modeling the spread of infections, Cooke et al. [9] used the following delay differential equation d N dt = r ( N ( t 7 ) ) N ( t ~') cN(t), where c > 0 and ~> 0 are suitable constants. Consider a logistic-like equation with a harvesting (hunting) strategy where we do not know the population N ( t ) at the exact time. However, we need (Example 1.2) it to determine the hunting quota. There is always a delay in processing and distributing field information. The quota must be set long before the hunting season begins. The time lags in ecological systems can be justified as a discovery time: predator requires time to discover the prey is very abundant. In the case of metapopulation (Example 1.1), time lags can be a result of the situation where prey population becomes sufficiently rare, such that the predator switches to alternative prey or an alternative source of prey (different island). It might be reasonable to consider a predation function as a function of the delayed estimate of the true population. It is important to be able to model this delay because it has serious implications for the long-term behavior of populations. Time lags yield specific insights into the management of complex ecological systems. It is considered that the major effect of delays is to make them less stable than the analogous models without delays. The introduction of delays into existing mathematical ecology equations is supported by general arguments that the interacting species somehow rely on resources and harvesting that have been accumulated in the past. It is now a well-known fact that small delays may cause some otherwise inexplicable periodic oscillations. Equation (1) with delays can be rewritten in the following form d N dt = r ( N ( h ( t ) ) , t ) N ( t ) c ( t ) N ( g ( t ) , t ) , h ( t ) <_ t, g ( t ) <_ t. (3) In this paper, we consider equations of type (3) with several delays in the logistic and the harvesting parts. Here we restrict ourselves to the linear harvesting function E = c ( t ) N ( t ) (Type I of response). Such functions were used in Examples 1.1 and 1.2. We will obtain here sufficient conditions for positiveness, boundedness, and extinction of solutions of equation (3). Similar problems for the equation with constant harvesting were studied in [12,13]. In [14,15], a general logistic delay equation without harvesting was investigated, a p r i o r i upper bounds of solutions were obtained and applied to derive explicit conditions of global stability for these equations. In the present paper, we also find a pr io r i upper bound of solutions for equations with harvesting and use this estimate to study the positiveness of solutions. Our theoretical results are illustrated by numerical simulations. The paper is organized as follows. Section 2 includes relevant results for linear and nonlinear delay differential equations. Section 3 contains main results on the existence of positive solutions which do not tend to zero. This corresponds to nonextinction of the population. Some estimates for positive solutions are also presented. In the end of Section 3, we apply these results to the metapopulation model and to the lobster fishery equation with delay (see Examples 1.1 and 1.2). In Section 4, the results are discussed and illustrated by numerical examples. 2. P R E L I M I N A R I E S Consider a scalar delay differential equation N ( t ) = r ( t ) N ( t ) a b k N ( h k ( t ) ) c l ( t ) N ( g z ( t ) ) , =1 =1 t _> 0, (4) 1246 L. BEREZANSKY et al. with the initial function and the initial value N ( t ) = ~ ( t ) , t < 0 , N ( 0 ) = N 0 , (5) under the following conditions: (al) a > 0 , bk >0; (a2) r(t) > O, cl(t) >_ 0 are Lebesgue measurable and locally essentially bounded functions; (a3) hk(t), gz(t) are Lebesgue measurable functions, hk(t) <_ t, gt(t) <_ t, limt-,oohk(t) = ce, limt_~o~ gz(t) = oo; (a4) T : (--oo, 0) --* R is a Borel measurable bounded function, ~(t) > 0, No > 0. DEFINITION. An absolutely continuous on each interval [0, b] function N : R --* R is called a solution of problem (4),(5), i f it satisfies equation (4), for almost all t E [0, oo), and equalities (5) for t <__ O. We will present here lemmas which will be used in the proof of the main results. Consider the linear delay differential equation n Jc(t) +]=1 cL(t)x(gl(t)) = O, t > 0 , (6) and a corresponding differential inequality n 9(t) +]=1 c~(t)y(g~(t)) _< o, t __ o. (7) We say that a function is nonoscillatory if it is eventually positive or eventually DEFINITION. negative. LEMMA 1. (See [16].) Suppose that for the functions ct, gt Hypotheses (a2) and (a3) hold. Then, (1) i f y ( t ) is a positive solution of (7) for t > to > O, then y(t) <_ z(t) , t > to, where z(t) is a solution of (6) and x(t) = y(t), t _< to, (2) for every nonoscillatory solution x(t) of (6) we have limt-.oo x(t) = O, (3) if °/m 1 sup ] c~(s) ds < , (8) t>0 ]=1 ink gk(t) ¢ then equation (6) has a nonoscillatory solution. I f in addition, 0 <_ ~(t) < No, then the solution of initial value problem (6),(5), where N(t) in (5) is replaced by x(t), is positive. Consider also the following linear delay equation with positive and negative coefficients t > o . (9) n ~:(t) + ]=~ cz(t)=(g~(t)) ~( t )=( t ) = o, DEFINITION. A solution X( t , s) of the problem ~(t) +]=~ cz(t)=(g~(t)) a(t)=(t) = o, = ( t ) = o, t < ~, =(~) = 1 t >s , is called a fundamental function of (9). Denote G(t) = maxz gl(t). Delay Different ia l Logis t ic E q u a t i o n s 1247 LEMMA 2. (See [17,18].) Suppose that for the functions cl,gz Conditions (a2) and (a3) hold, a is a locally bounded function, a(t) > O, ct(t) > a(t), cl(t) a(t) dt = oo, (iO) =1 1