Continuous versus discrete single species population models with adjustable reproductive strategies

Bulktin of MaIhematical Biology, Vol. 59, No. 4, pp. 619-705, I991

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0 1997 Society for Mathematical Biology

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CONTINUOUS VERSUS DISCRETE SINGLE SPECIES POPULATION MODELS WITH ADJUSTABLE REPRODUCI’IVE STRATEGIES

MATS GYLLENBERG Department of Mathematics, University of Turku, FIN-20014 Turku, Finland

(Email: [email protected])

ILKKA HANSKI Department of Ecology and Systematics, University of Helsinki, FIN-00014 Helsinki, Finland (Email: [email protected])

TORSTEN LINDSTRijM* Department of Technology and Natural Sciences, University of Orebro, S-70182 &ebro, Sweden

(Email: [email protected])

We investigate population models with both continuous and discrete elements. Birth is assumed to occur at discrete instants of time whereas death and competition for resources and space occur continuously during the season. We compare the dynamics of such discrete-continuous hybrid models with the dynamics of purely discrete models where within-season mortality and competition are modelled directly as discrete events. We show that non-monotone discrete single-species maps cannot be derived from unstructured competition processes. This result is well known in the case of fixed reproductive strategies and our results extend this to the case of adjustable reproductive strategies. It is also shown that the most commonly used non-monotone discrete maps can be derived from structured competition processes. 0 1997 Society for Mathematical Biology

1. Introduction. Mathematical models describing the dynamics of a population can be classified according to whether time is considered as a discrete or continuous variable. The growth of a semelparous population reproducing at discrete instants is often most naturally modelled by a

*Present address: University of Oslo, Department of Biology, Division of Zoology, P.O. Box 1050, Blindern, N-0316 Oslo, Norway.

679

680 M. GYJLENBERG et al.

difference equation. On the other hand, if generations overlap and births are distributed over the year, a continuous model based on differential equations is usually more appropriate. In reality, population dynamics usually involves some continuous processes and some discrete processes. For example, reproduction often occurs in short breeding seasons but competition and death occur continuously. The following question is therefore of interest: Under which conditions does a discrete model successfully capture the consequences of continuous intraspecific competition during the season?

We want to stress that it is not obvious how discrete and continuous models are related to each other. Several authors simply, but incorrectly, derive discrete analogues to continuous models by replacing differentials by differences. In the single-species case the best known example is the continuous logistic equation

n(t) h(t) =pn(t) 1 -K , ( 1

which leads to the discrete quadratic map

N’=PN 1-g , ( I (2)

if differentials are replaced by differences. (In equation (l), n(t) is the population density at time t and the dot denotes differentiation with respect to time. In equation (2), N is the population density at a given instant of time and N’ the population density one time unit later.) Because of this derivation, equation (2) is often referred to as the discrete logistic map (May, 1973; Cooke and Witten, 1986). Unfortunately, this type of discretization may lead to biologically meaningless models which, for instance, predict negative numbers of individuals. This is indeed the case with equation (2) if p > 4 or if the initial population size exceeds K.

Maps other than the one given by equation (2) are also referred to as discrete logistic maps. For instance Allen et al. (1993) and Pielou (1977) call the map

AN N’ =

l+(A-l)N/K’

with A = exp( p), the discrete logistic map. The derivation in Pielou (1977) shows that the solution of (3) gives the solution of the continuous logistic equation (1) at discrete time instants.

SINGLE SPECIES POPULATION MODELS 681

The maps (2) and (3) differ considerably from each other dynamically. All non-trivial solution to (3) converge monotonically to a unique non-trivial fixed point. On the other hand it is well known that the map (2) exhibits extremely complicated or “chaotic” behavior for p large enough (May, 1974a, 1981; Murray, 1989 and the references therein).

It is the purpose of this paper to investigate population models with both discrete and continuous elements. Birth will be assumed to occur at discrete and equally distributed instants, while death and competition for resources and space occur continuously during the season between reproductive events. Solving the within-season continuous model we arrive at a discrete map taking the population state immediately after a reproductive event to the population state immediately after the following reproductive event. We shall compare the dynamics of such discrete-continuous hybrid models with the dynamics of purely discrete models where within-season mortality and competition are modelled directly as discrete events. In particular, we want to understand what kind of individual behavior may lead to non-monotone maps like the map (2).

The within-season competition is a non-linear process: the survival probability of an individual depends on the outside world through so-called environmental interaction variables (see Diekmann et al., 1997), which in turn are influenced by the activity of the population. Similarly individual reproduction is affected by the environmental interaction variables. Breed- ing females of many animals are capable of responding to changing environmental conditions by for instance reducing clutch size, resorbing embryos and possibly by selectively lactating only some of their offspring. We shall distinguish between fixed and adjustable reproductive strategies. By a fixed reproductive strategy we understand a model where an individual at each reproductive event produces on average a fixed number p of young. In an adjustable strategy p depends on the environmental interaction variable. We shall consider competition between different reproductive strategies. The fundamental question is of course that of invasability of a resident type by a type following a different strategy.

Throughout the paper we shall take the total population size as the (single) environmental interaction variable. If one wants to stay within the realm of unstructured, single-species models, this is essentially the only possible choice.

It is well known that with a fixed reproductive strategy unstructured continuous death processes (that is, processes modelled by a single ordinary differential equation) cannot give rise to unimodal discrete maps, such as (2) or the Ricker (1954) model

N’=PNexp(-kN). (4)

682 M. GYLLENBERG et al.

Because of its importance for our argumentation, we repeat in section 2.1 the simple proof of this fact much in the spirit of Beverton and Holt (1957), Clark (1990) and Yodzis (1989). It is clear that adjustable reproductive strategies may very well lead to unimodal maps, but in section 2.4 we present one of our main results, namely, that in the context of our models reproductive strategies leading to non-monotone maps are not optimal but can be outcompeted by reproductive strategies leading to monotone maps. In this sense unstructured continuous within-season competition models cannot account for evolutionarily stable non-monotone discrete models in one variable. In order to get reasonable mechanistic explanations of such non-monotone discrete maps we therefore have to consider structured population dynamics in the sense of Metz and Diekmann (1986).

Some derivations of models like (4) with reference to population structure have already been made (Clark, 1990; Hanski, 1989; Ricker, 1954; Yodzis, 1989). The derivation in (Clark, 1990; Ricker, 1954; Yodzis, 1989) is based on the observation that adult fish cannibalize their own offspring. Another structured population dynamical explanation of Ricker-type models was proposed by Hanski (1989). He divided the population into floaters, residents and dispersers. The floaters represent mature individuals searching for reproduction opportunities. The residents are reproducing individuals and dispersers are individuals that have surrendered their reproduction opportunities. The numerical results by Hanski (1989) show that discrete models close to the Ricker model (4) can be derived from a population structure based on floaters, residents and dispersers.

The need for structured approaches when deriving non-monotone discrete maps is closely related to the notion of “scramble” and “contest” competition (Hassell, 1976; Nicholson, 1954). Scramble competition can be viewed as the case when resources are extremely well distributed and there is a perfect sharing of resources among the individuals. Either all individuals survive and succeed to reproduce or the population goes extinct. This is often a good approximation in cases when ephemeral resources like car- rion, dung and stored products are exploited (Hassell, 1976).

In contrast, in contest competition some individuals get all they require, while others may have insufficient resources for survival or reproduction. Contest competition occurs when the individuals compete for a fixed number of refuges, territories or a position in a social hierarchy (Hassell, 1976). An unstructured continuous competition model can only take into account the number of winners in a contest-type competition. Hence, it describes competition best when the losers are removed from the population or the losers do not affect the dynamics of the winners. However, this may not be the case. For example, Maynard Smith and Price (1973) and Maynard Smith (1974b) suggest that intraspecific competition does not cause severe injury or death, and hence the losers remain in the population and, if they are many, they may substantially affect the dynamics. In section


3 of this paper we analyze a simplified version of the structured floater- resident-disperser model introduced by Hanski (1989). Our approach contains the pure contest-type, the pure scramble-type and the Ricker model as special cases of a wide range of discrete models. We shall study the effects of adjustable reproductive strategies in these models and find that most of the regularly used discrete models (May and Oster, 1976; May, 1981) can be put into the general framework presented here.

2. Unstructured Models. Unstructured population models are based on the assumption that all individuals experience the same risk of dying and on average give birth to the same number of offspring. Survival and reproduction are, however, not constant, but are affected by intraspecific competition. We shall assume that competition takes place through the environment: The population affects its environment, which in turn determines the survival probability and reproduction intensity.

The growth of a semelparous, single-type population reproducing at discrete instances can under these assumption be described by the model

N’ =s?(E)N, (5)

where N is the population size and E is the value of the environmental interaction variable (in the sequel called simply environment for short) at the beginning of a season (immediately after reproduction) and N’ denotes the population size at the beginning of the following season. L%?(E) is the basic reproduction number, that is, the expected number of offspring born to an individual. In general 3 can be written as a product

am = pmmm, (6)

where F(E) is the probability that an individual survives the season (in the sense that it gets the opportunity to reproduce) and P(E) is the expected number of offspring per surviving individual when the environment at the beginning of the season has the value E. As pointed out in the Introduction we shall always take the total population size as the environment. In the single-type case we thus choose

E=N. (7)

Assume now that the survival probability F in the model (5)~(7) is determined by intraspecific competition occurring continuously during the season. Assume furthermore that competition acts in such a way that the per capita death rate at time t is a function ,u(e(t)) of the environment e(t) at that particular instant of time. The size of the population between two

684 .M. GYLLENBERG et al.

reproduction instants thus satisfies the differential equation

ri(t) = -pL(e(t))n(t), O<t<T,

n(0) =N, (8)

where N is the population size immediately after the previous reproduction instant and T denotes the length of the season. The feedback relation analogous to (7) is

e(t) =n(t>. (9

Solving-the initial value problem (8) and (9) one obtains the population size n(t; N) as a function of time and initial state after which F is obtained from F= n(T; N)/N.

In this section we shall investigate the competition between types playing different strategies, that is, types having different L% and ps. We empha- size that since competition is assumed to take place through the environment, all types experience the same E and e(t). We shall restrict our attention to models where the environment is given by the total population size.

In section 2.1 we derive the discrete model (3) from a model with continuous within-season competition. In section 2.2 we consider competition between two fixed strategies with continuous within-season competition. In contrast to the single-type case, the corresponding discrete system cannot be written down explicitly. We therefore derive and analyze an approximation of this discrete system in section 2.3. In section 2.4 and 2.5 we consider adjustable reproductive strategies and pay special attention to the differences in the dynamical behavior between models derived from continuous within-season competition and models where the competition contains a discrete element.

2.1. The Bevetion-Halt model. The solution to the initial value problem (8) is given by

n(t; N) = H-‘(H(N) -t), (10)

where H is defined by

H(x)=/;‘&. (11)

Notice that if Al. is positive, H is well defined by (11) and increasing. It follows that the solution (10) of (8) is an increasing function of N. As a consequence, if p is constant, the map NH N’ defined by (5)-(7) by

sT(N) = H-‘(H(N) -T) N

(12)

is monotonically increasing.


Example 1. The right-hand side of the continuous logistic equation (1) is occasionally interpreted as the difference of a density-independent birth rate @z and a density-dependent death rate p(n)n with the per capita death rate

p.(n) = j.uz := ( P/K>n. (13)

Inserting (13) into (11) one obtains from (12)

(14)

We shall refer to survival probabilities of the form (14) as logistic survival probabilities. The survival probability (14) yields the following discrete version of the continuous logistic equation:

(15)

This derivation of (15) was carried out (in a more general context) by Beverton and Holt (1957).

If we put p = P(N) = exp( -kN)(l + @.‘N) in (15) we arrive at the Ricker model, but we show in Proposition 4 that such reproductive strategies are not likely to be long lived.

2.2. Continuous within-season competition between two fixed strategies. Adopting the terminology of Gyllenberg et al. (1996), we shall refer to a fixed reproductive strategy whenever p is constant. If /3 depends on E in some way, we refer to an adjustable reproductive strategy. Gyllenberg et al. (1996) investigated the dynamical consequences of adjustable strategies and found that adjustable strategies either stabilize or outcompete fixed strategies. Moreover, if the adjustable strategy outcompetes the fixed strategy, then the oscillations have smaller mean amplitudes in the adjustable strategy case than in the fixed strategy case.

We first determine the best fixed strategy and then compare it with the best adjustable strategy. We consider competition between two asexually reproducing types N and P of the same species. The total population size is thus given by e(t) = n(t) +p(t). Assuming logistic competition, the within-season dynamics is described by

h(t) = -pIe(t Q<t<T,

$(t) = -j.k2e(t)p(t), O<t<T,

e(t) = n(t) +p(t),

A(0) = N,

pm = P,

(16)

686 M. GYLLENBERG etal.

where N and P are the population sixes of the two types immediately after the previous reproduction instant. The coefficients Jo,, p2 reflect the competitive ability of the different types; the one with the smaller p has the greater probability of survival. The assumption of fixed reproductive strategies yields the discrete model

N’ = &n(T; IV, P), (17)

P’ = &P(T; N, P),

where (n(T; N, P), p(T; N, P)) is the solution of the initial value problem (16).

PROPOSITION 1. Consider the system (16) and (17) with pl, kz positive and PI, Pz > 1. If

1% P2 1% PI -<- (18)

P2 I4 ’

then the point (( PI - l)/(Tpi), 0) attracts every point not on the P-axis. If the inequality in (18) is in the reverse direction, then (0, ( p2 - l)/(T&) attracts every point not on the N-axis.

Proof: The orbit of (16) lies on the curve

Condition (18) says precisely that at each reproduction instant the state (N, P) of the system jumps to a curve of type (19) which is closer to the N-axis than the curve on which the state moved during the previous season. Since ( pi - l)/(Tpi) attracts every point on the N-axis except the origin, the conclusion follows.

Interchanging the roles of N and P one obtains the result in case the reverse inequality holds in (18). ??

2.3. A discrete model of competition between two fixed strategies. In general the initial value problem (16) cannot be solved in closed form and hence one cannot write down the corresponding discrete dynamical system (17) explicitly. As shown in Example 1 the discrete system takes the form (15) on the coordinate axes. It is therefore tempting to approximate the system (16) and (17) by a two-dimensional version of (15) with the survival probability (14) depending on total population E = N + P. We thus con-


sider the system

N’ = &qE)N,

P’ = /3pgEP,

E=N+P,

with

1 T(E) =

1 + /.QTE ’

(20)

Cm

Gyllenberg et al. (1996) showed that if competition is described by the Ricker model, that is, if the survival functions are given by

5$(E) =exp(-kE), (22)

then the type with the largest equilibrium population size outcompetes all other types regardless of the stability of this equilibrium. We now prove the corresponding result for the case where the function aE)E are monotone functions of the total population size E. In this case the one-dimensional dynamics of one type in the absence of the other is, of course, very simple with all solutions except the trivial one approaching the steady state monotonically.

PROPOSITION 2. Consider the system (20) and assume that

(A-I) q is positive, decreasing and continuous for i = 1,2; (A-II) E ++$E)E is increasing for i = 1,2;

(A-III) T(O) = 1, S$(m) = 0, i = 1,2.

Let &, p2 > 1 and let N* and P* be the unique solutions of the equation P&(N) = 1 and &FJP) = 1, respectiveZy. Then (N*,O) and (O,P*) arefixed points. If N* > P*, then the N+xis attracts evey point not on the P-axis and if N* < P*, then the P-axis attracts every point not on the N-axis.

Prooj It is clear that (N*, 0) and (0, P*) are fixed points. Assume that N * > P* (the proof for the case in which the reverse inequality holds is identical). Consider the curves yi = pin E) in the Ey-plane. By (A-I) and (A-III) there exists a number 6 > 0 such that the curve y, is strictly above y, for P* - S <E <N* + 8. It follows that there exists a k > 1 such that

N’ N p-kp (23)

688 M. GYLLENF3ERG et al.

for all points (N, P) in the strip

cn={(N,P): p*- 6<E<N*+iY,N>O,P>O}. (24

Clearly, for every point (N, P) in the interior of the positive quadrant, the total population E’ = N’ + P’ in the next generation lies between G,(E) := &9JE)E and G,(E) := P&(E)E. However, by (A-II) both the curves yi = Gj( E), i = 1,2, lie between the broken lines

if E<P*, , if P* <E,

and

y,= ;*'

i

ifE<N*, , if N* <E,

in the Ey-plane. It follows that the set fi defined by (24) is invariant and that every orbit starting in the interior of the positive quadrant enter n in a finite number of steps. The relation (23) now implies that such an orbit converges to (N *, 0). ??

Our results are concerned with the global dynamical behavior of competition systems and they give conditions for competitive exclusion where the winning type actually drives the other type to extinction. In particular, the assumptions ensure successful invasion of the winning type into a population consisting of individuals of the other type.

Our models have the special feature that the environment E (the total population size) is a one-dimensional variable. Moreover, by assumption (A-I) the basic reproduction numbers of the types are decreasing functions of E. Diekmann and Mylius (1995) showed that it follows from this that if population dynamics leads to a steady state with constant E, then a type x is evolutionarily stable if and only if the function which assigns to a type the environmental value at which the population will be steady has a maximum at n. That selection under suitable conditions tends to maximize equilibrium population size was pointed out already by Fisher (1930). We refer the reader to the book by Charlesworth (1980) for a detailed discussion of this principle and the conditions under which it is valid. Charlesworth (1980) also gave an account of an experiment where the population did not settle down to a stationary population size, but where the average population size increased as a consequence of selection.

The logistic survival functions (21) obviously satisfy the assumptions of Proposition 2, and for the rest of this section we shall restrict our attention to such survival functions.

We now compare the dynamics of the system (16) and (17) with that of its approximation (20) and (211, especially with respect to Fisher’s principle alluded to above.


In the system (20) and (21) the type with the larger equilibrium density will indeed outcompete the other type. In the model (16) and (17) the situation is different. An equilibrium of the discrete dynamical system corresponds to a stationary population size on the generation level only and between two reproductive events the population size e(t) changes due to competition described by the continuous system (16). The type with the larger equilibrium density may very well lose competition (one can easily find parameters &, &, pi, p2 satisfying both (18) and ( /3i - l>/pi < ( p2 - 1)/p2). Notice, however, that in the absence of the other type the average population density when the discrete system (17) is at equilibrium is log p/p. Proposition 1 therefore shows that in the context of system (16) and (17) the type with the largest average population density outcompetes the other type.

2.4. Adjustable strategies in the discrete competition model. We consider system (20) and (21) and assume a trade-off between reproductive effort and the competitive ability by making k a function of p. We assume that offspring are produced in clutches, that larger values of /3 correspond to larger clutches and that individuals produced in small clutches do better in competition than individuals produced in large clutches. This assumption has substantial empirical support for many organisms (Stearns, 1992) and requires p to be an increasing function of p. (We use the term “clutch” as a shorthand for a batch of offspring born at the same time, with no intention of restricting the results to species for which clutch is generally used in the biological literature.) In reality the constraint between /3 and p probably quite often depends on the environmental interaction variable, but in our setting an environment-dependent constraint would exclude fixed strategies, because p would then automatically depend on the environmental interaction variable. Since we are interested in competition between fixed and adjustable strategies we therefore assume that the trade-off does not depend on the environment.

Proposition 2 implies that the type with the largest equilibrium density in the absence of the other type will outcompete the other. Since this equilibrium is given by

P-1 h(P)= Tp(p) 9 (25)

we infer that the type with the largest h( p) will win competition described by (20) and (21). For physiological reasons individuals cannot produce an arbitrarily large number of offspring. We denote the upper bound of the clutch size by M. It teems reasonable to assume the existence of a unique optimal clutch size p which is strictly less than the physiologically deter-


mined maximal clutch size M. Mathematically this amounts toArequiring that the function h defined by (25) has a unique global maximum p E (0, M). In natural populations it may very well be that the maximal clutch size depends on the value of the environmental interaction variable. However, making M a function of E one would exclude fixed strategies and therefore we assume that A4 is a given constant.

We make the following assumptions about CL:

(B-I) p: (0, M] + R, is positive and increasing; (B-II) the function p * (/3 - l)/(Tp( /3)) has a global maximum 6~

(0, M); (B-III) or. E C2(0, 44) and p”( /3) > 0 for all p E (0, M).

Assumption (B-III) is needed to ensure that the optimal clutch size b provided by (B-II) is indeed unique. The three assumptions are therefore, as explained above, biologically justified. Notice that the function p( p> = pP, p > 1, satisfies (B-I)-(B-III) and this p will serve as our prototype in the sequel.

The be$ fixed reproductive strategy in the model (20) and (21) is to produce /3 offspring on average. We now consider a type P able to adjust its clutch size optimally according to the environmental conditions. By this we mean that, given the state E of the environment, the individuals adjust the average number of offspring to maximize the basic reproduction number s(E). Assuming a logistic survival function (21) we obtain the discrete dynamical system

P’ =P sup P

O<P<M 1+ QJ( P)E (26)

where E = P. Gyllenberg et al. (1996) proved an analogue of the following proposition

in the case of survival functions of Ricker type (22). Since the proof is similar, we do not repeat it here.

PROPOSITION 3. Assume (B-I)-(B-III). Then

(9

(ii)

The supremum in (26) is a maximum and for every E E R, it is attained at precisety one j3 E (0, Ml. The unique p =: p(E) found in (i) defines a continuous function 6: R, + (0, M] which takes on the value M in

IO, l/(M&UI - &I)T)I,

is strictly decreasing in

[l/(M/J(M)T - /.&I)T), 00)


and satisfies

s. (27)

(iii) Put

1

M/J(M)T - &M)T

The map P H P’ defined by (26) is continuously difirentiable on R+\_Y.

Conclusion (ii) of Proposition 3 says that at low densities the type P playing the adjustable strategy produces the largest physiologically possible clutch. As the population density E increases above a threshold value l/(Mp’(M)T - p(M)T), this type starts to suppress reproduction. If the population density is larger than the equilibrium density ( p - l)/Tp( p), then the type playing the adjustable strategy will actually produce smaller clutches than the type N playing the best fixed strategy.

Example 2. We compare hxed and adjustable strategies whe? p.( /3 > = /3 p, p > 1. The fittest fixed strategy corresponds to the clutch size p = p/( p - 1) and its dynamics are described by the map

P N

N’= P-I I+(p/(p-1))‘TN’ (28)

In order to calculate the corresponding map for the adjustable strategy, we note that /? = ’ l/(P( p - 1)T) and p( p> = l/(P( p - 1)T). Thus the dynamics of the adjustable strategy is described by the map

p’ = ,

I

PP 1 P-I i

P-1 _p(p- P

+ (+)‘TPi’, P<

P/ 1 -1)/p

\i (p-‘1)T7 p>

1

(p-1)WT’ (29

1

(p-1)MT’

We note that the adjustable strategy has no dynamical consequences, since each of the maps (28) and (29) has a globally asymptotically stable fixed point. However, the asymptotic differences of these maps are considerable. The map corresponding to the fured strategy is bounded, but the map corresponding to the adjustable strategy tends to infinity as the population size tends to infinity.


The simple argument in section 2.1 showed that fixed reproductive strategies combined with continuous within-season competition always give rise to monotonically increasing discrete maps. Example 2 showed that the map (26) describing the dynamics of a type playing the optimal strategy under logistic survival is increasing in the case p( /3) = PP. As a matter of fact, this result is not restricted to the model (26), but holds quite generally and in particular whenever the conclusions of Proposition 3 hold. This is the content of the following proposition, which is so obvious that it requires no proof.

PROPOSITION 4. Let f: [O,w)’ + [0, QJ) be continuous and assume that for all /3 E (0, M] the function g(N) = f( p, N) has the following properties:

(a) Ng(N) i+s strictly increasing in N > 0; (b) g(N) is strictly decreasing in N > 0.

Then the function

g(N) = sup f( /3, N) O<B<M

has the properties (a) and (b) too.

The biological implication of Proposition 4 is that if the population dynamics of a single type is described by a non-monotone map, then the underlying reproductive strategy does not maximize the net reproduction number-it is not optimal.

Next we investigate competition between the best fixed strategy and the adjustable strategy.

According to Proposition 3(ii)_ the model (26) for the type P can be written in terms of the function P(E). Choosing the total population as the environmental interaction variable, we obtain the following model for the competition between the best fked and adjustable strategies:

N’= l+Tp(@E'

,&E)P ” = 1 + T/J( &E>)E’

E=N+P.

(30)

We have the following result, the proof of which is similar to an analogous result in Gyllenberg et al. (1996) and therefore is omitted.


PROPOSITION 5. Assume (B-I)-(B-III). Every point OIZ the segment

N+P= p-1

CL( Ei)T’ N z 0, P > 0, (31)

is a (Lyapunov) stable jked point of (30) which is not asymptotically stable. For eve9 initial state Q E Int R:, o(Q) consists of a single point on the segment (31) and depends on the initial state Q. One has

P’ P -a- N’ N

(32)

with equality holding if and only if (N, P> is on the segment (31).

According to Proposition 5 every orbit starting in the interior of the positive quadrant approaches a neutrally stable equilibrium on the segment (31). AS long as the state (N, P) of the system has not reached this segment, it moves in the direction of the P-axis. This implies that if an equilibrium state is perturbed such that both the N and P types are affected equally, the system will settle to a new equilibrium which is closer to the P-axis than the previous equilibrium state. Since real systems are always subject to random perturbations, the adjustable strategy will eventu- ally outcompete the best fixed strategy.

2.5. Adjustable strategies in the continuous within-season competition model. The discrete model (30) assumes logistic survival function and is not derived from continuously occurring within-season competition. As noted above in the case of competition between two fixed reproductive strategies (Propositions 1 and 2), this assumption can make a real difference. Next we shall therefore analyze competition between fixed and adjustable strategies, assuming it takes place continuously during the season. Our results are not as complete as in the case of fixed strategies, but we show that under reasonable assumptions an adjustable strategy can indeed invade a population consisting of individuals playing the best fixed strategy.

According to Proposition 1 the best fixed reproductive strategy when the within-season competition occurs continuously is obtained by maximizing log P/PC p). Our assumptions (B-I)-(B-III) imply that such a best tied strategy actually exists, that is, that the function p e log /3/~( /3> has a unique global maximum p. In fact more is true: the function P I+ log P/LL( P) is unimodal and p is strictly less than p = max{( p - l)/Tp( /3)), the birth ratio corresponding to the best fixed strategy with logistic survival function (recall Proposition 2). This is the content of the following lemma.


+EM~ 1. Let (B-I)-(B-III) hold. Then p e log p/p( /3) k unimoahl and P<P*

Proof Put i( /? > = ,u( p > - ( /3 - l)p’( p 1. The sign of h equals the sign of the derivative of ( p - l)/(Tp( p)). (B-I) gives h(O) = ~(0) + k’(O) 2 0. Since fi’( p > = ( p - 1)~” ( /3 ), (B-III) gives that fi increases on [0, 11 and decreases on [l, Ml. That is, i has at most one zero on (0, M]. (B-II) implies that this zero exists and is located in (1, M). This” zero is 6. Similarly, we put h( p) = p( p) - /3 log &.L’( /3). The sign ofyh( p) equals the sign of the derivative of log /3/~( p). (B-I) gives h(O) = ~(0) - O*E.L(O) 2 c. Since h’( /?> = -log p( p’( P) + pp”( P>), 03-I) and (B-111) give that h( p) increases on [0, 11 and decreases on [l, Ml. This implies that i( p> has at mFst one zero in (0, M] and this zero is, if it exists, 1ocaJed in (1, Ml. Now, h( /?) - h( /3) =p’( p)( /3 log p - ( /3 - 1)). We have h(l) - h(1) = 0. AMvreovzr,fifi’( p) - h:( /3) = iog p > 0, if p E (1, M]. This implies that 0 = h( p) > h( p). Hence J3 E (1, p). w

Recall that in the discrete competition model the best adjustable strategy was to suppress reproduction for high population densities. The birth ratio p(E) maximizing the net reproduction number was given in Proposition 3(G). We now define an adjustable reproductive strategy by

p(E) = s7 E Q I/( h’( sp- l.L( I+)9 fi(E), E ’ l/( /%‘( @T - p( j)T).

(33)

For low population densities E the strategy is to produce clutches corresponding to the best fixed strategy and as the population density grows, the type starts to suppress reproduction as in the case of logistic survival functions. We do not claim that strategy (33) is the best adjustable strategy, but it is a reasonable one that under the assumption of suitable perturba- tion can invade a population playing the best fixed strategy.

Assuming (331, within-season competition is described by the system

rib) = -j~( /i)ett)n(t), O<t<T,

b(t) = - p( p(E))e(t)p(t), O<t,<T,

e(t) =n(t> +p(t), (34) n(0) = N,

p(0) = P,

e(0) = E,


resulting in the discrete dynamical system

N’ = jn(T; N, P),

PI = p(E)p(~; N, P),

E=N+P.

(35)

We have the following result.

PROPOSITION 6. Assume that (B-I)-(B-III). Every point on the segment

N+P= j-1

F( PI)T ’ N > 0, P > 0, (36)

is a (Lyapunov) stablefixedpoint of (34) and (35) which is not asymptotically stable and for every initial state Q E Int R:, w(Q) consists of a single point on the segment (36). One has

P’ P N’+ (37)

if E> l/(&.~'(p)T- p(b)T), and

P’ P N’=$ (38)

otherwise. The exact location of w(Q) depends on the initial state Q.

Proof We observe first that if

1

E’ &..@)T-@)T (39)

we get p(E) = B from (33) and the equality (38) follows. In the case

1 N+P>

k( @T- p( @T (40)

the idea is to compare the solutions of (34) and (35) to the solutions of

N’ = fiN

l+T@)E

p’ = fi(E)P

1+ Tp( P(E))E’

E=N+P.

6% M. GYLLJHBERG et a!

Put

N n, 0) =

1 +t,( @(N+P) ’

P p*(t) =

1 +t,( @(N+P)

and let n(t), p(t) be the solution to (34) for 0 G t G T. We have

N+P N+P

l+tp(j)(N+P) <n(t) +p(t) <

1 +tp( B)(N+P)

and

ri*w h(t) n,(t)

> -P( @(n(t) +pW = n(t>,

@*Cd @W

P*(t) < -cL( B)Mt) +pm = -

p(t) ’

so, consequently, n,(t) > n(t) and p*(t) <p(t). It follows that

PN P’ BP* CT) Sup v l+T&3)(N+P) = O<B<P P N” v

pn* (T) SN ‘3

l+Tp(@N+P)

??

As the proposition shows, the adjustable strategy outcompetes the fixed strategy if the system is subject to large perturbations affecting both types evenly. These perturbations must give rise to population densities considerably higher than the equilibrium density. Proposition 6 demonstrates that adjustable strategies can invade, but that the probability for this may be small.

3. A Structured Model. In the preceding section we showed that, within the context of our class of unstructured single-species population models with discrete reproduction and continuous within-season competition, fixed reproductive strategies cannot give rise to unimodal maps and that adjustable strategies leading to unimodal maps are not optimal. On the other hand, discrete-time population models described by unimodal maps like the


ticker model have proved very useful in describing and explaining population fluctuations in nature. In this section we shall therefore show that by introducing population structure, one can arrive at unimodal maps.

We consider a model describing competition for a place to reproduce. Our basic assumption is that intraspecific competition is not lethal. The losers in this competition are therefore not removed from the population, but they cannot reproduce. Individuals not able to reproduce (flouters) remain in the population and compete for resources with reproducing individuals (resLfe&s) during the reproductive season. This assumption has substantial empirical support (Maynard Smith and Price, 1973; Maynard Smith, 1974b; and references therein).

The model is structured in the sense that individuals are classified as either floaters or residents. All individuals are born as floaters. These individuals are searching for a place to reproduce. As they find suitable breeding territories, they become residents. The actual competition for breeding territories is modeled by assuming that floaters become residents at a per capita rate proportional to the available space. We assume that each resident occupies a space of constant size which is proportional to the average clutch size p. We thus assume that the size of the territory which a resident individual is willing to defend increases with her clutch size, because the value of the extra space is greater for individuals with greater clutch size (Milinski and Parker, 1991).

As a consequence, there is a maximum value, determined by the total area, for the number of individuals which may become residents and reproduce. On the other hand, floaters are able to squeeze so that the whole space may contain an arbitrarily large number of floaters. For simplicity we neglect mortality during the season. We therefore arrive at the following continuous model for the within-season dynamics:

f(t) = -fW(l -gGfW) - apm), O<t<T,

i(t) =f(t>(l -g(&fW) - aprw>, O<t<T, (411

f(O) = N,

r(O) = 0.

Here f(t) is the density of floaters and r(t) is the density of residents at time t. a is a proportionality constant. The parameter k is the area occupied by one floater at low density and g(kF) is the fraction of the whole space that is occupied by floaters when the density of floaters is F. With this interpretation it is clear that g has to be non-decreasing, bounded above by 1 and satisfy g(O) = 0 and g’(O) = 1. In addition we assume that it is continuous. We have chosen the time unit such that the per capita rate at which floaters become residents at low population densities equals 1.


Since we assume semelparity, the floaters still remaining at the end of the season will be removed. Combined with the assumption of a fixed reproductive strategy, the model (41) therefore gives rise to a one-dimensional dynamical system

N’ = &F(N)N (42)

of type (5)-(7) with survival function

-eNI = t(T)/N, (43)

where r(t) is the second component of the solution of the initial value problem (41). We shall approximate the system (42) first under the assump- toin that the season is very short (T --) 0) and then under the assumption of a very long period (T + co) between two reproductive events. After this we integrate the system (41) numerically for two typical choices of g and certain values of T E (0, w) to obtain a graphical representation of the one-dimensional dynamical system (42). These results are compared with the extreme cases T = 0 and T = 00.

If T is small, we can employ the usual Euler which gives r(T) = TN(1 - g( kN)) and hence

difference approximation

N’ = pTN(1 -g(M)). (44)

If g is a concave function increasing sufficiently rapidly to 1 as N tends to infinity, then the map (44) is a unimodal map resembling very much the Ricker curve (4) or the quadratic map (2). As a matter of fact, it is possible to achieve exactly these maps by choosing the function g appropriately. Obviously the choice

g(N) = 1 - exp( -N) (45)

yields the Ricker model (4). Since this function is quite arbitrary, we cannot claim that we have given the Ricker model a mechanistic explanation. Consider now the function

g(N) = N-c 1, N> 1. (46)

The function (46) has a clear advantage over (45) since it can be given a mechanistic interpretation in terms of competition for territory: floaters require as much space as residents. The assumption is therefore that of


pure scramble competition (Hassell, 1976). It results in the dynamical system

N, = PTN(l -kNL

i

O<N< l/k, 0, N > l/k,

which is simply the quadratic map (2). Finally we notice that if floaters do not require any space at all and

hence do not affect the rate at which floaters become residents (pure contest competition (Hassell, 1976)), one has to choose g identically equal to zero and the resulting dynamical system is linear, predicting that the population either increases or decreases in a geometrical progression. This result contradicts the usual stabilizing effect of pure contest competition. The reason is that our assumption of a very short season is incompatible with the biological interpretation of pure contest competition. Biologically, pure contest competition should be resident driven in the sense that individuals who have already found suitable breeding territories inhibit floaters from becoming resident. However, if the season T is short, then according to the model (41), residents will remain scarce and their influ- ence on the dynamics will be marginal. Hence the dynamics is determined by the floaters and we have reached a contradiction. On the other hand, we show below that for large values of T, the choice g = 0 yields the usual predictions of pure contest competition.

If T is very large, r(T) can be approximated by lim, em r(t). We therefore investigate the asymptotic behavior of the system (41). The orbit obviously follows the straight line r +f = N. All points on the r-axis f = 0 and on the curve r = (1 -g(kf))/(ap) are equilibrium points and there are no others. For N sufficiently small, the straight line r +f = N lies entirely below the curve r = (1 - g(v))/(ap> and the solution approaches the point (0, N) as T + 03. There exists a critical value NC such that the straight line r +f= N and the curve r = (1 -g(w)>/(aj3> intersect if N 2 NC. In this case, the solution approaches the first point of intersection between the straight line and the curve. We denote the r-coordinate of this intersection point by q(N) (see Figure 1) and the system (42) and (43) takes the form

(47)

Note that the map (47) is in general not continuous at NC. Pure contest competition with g identically equal to zero now results in a

population growing geometrically until it hits the carrying capacity 1, after which the population density remains at the carrying capacity. Pure scramble competition with g defined by (46) also predicts an initially geometrically growing population which goes extinct the first time it overshoots the carrying capacity.

700 M. GYLLENESERG &al.

Na ‘.’ F

Figure 1. Phase diagram of system (41) with g(N) = 1 - exp(-N), k = 3/2, aB=l,T=m.

Figure 2 shows the number r(T) of residents at the end of the season as a function of the number N of individuals at the beginning of the season for different values of the season length T for our two main choices (45) and (46) of the function g. The curves were obtained by integrating the system (41) numerically. Because of the conservation law f(t) + r(t) = N, the system reduces to a scalar first order separable differential equation, but for g given by (45), the solution cannot be written in closed form. We notice that the curves resemble the Ricker curve and the quadratic map, respectively. For other choices of the concave increasing function g one obtains very similar results. We point out that even if the functions obtained using the approximations explained above give rise to quantita- tively different asymptotic behavior than the functions obtained by integrating the system (41), they still have the same universal features like period doubling route to chaos, etc.

We close this section by noting that one can easily combine the model (41) with an adjustable reproductive strategy. As in the unstructured models discussed in section 2, we assume a trade-off between reproduction and competitive ability during the season. We thus assume that the floaters have larger requirements for a habitat whenever they want to produce


0.B i)‘T=CQ

0.8

t II

0 1 2 N 3 0 0.5 1 1.5 N 2 (4 (b) -.

Figure 2. The number R(T) of residents at the end of the season as a function of the number of individuals at the beginning of the season for T = 1, 3, 10 and w. (a) g(N) = 1 - exp(-N), k = 3/2, up = 1; (b) g(F) = min(1, F), k = 1, up= 1.

R(T)

0.:~

larger clutches and, hence, they become residents more slowly in this case. This is taken into account by letting k be an increasing function of P:

f(t) = -f(Nl -g(k( P)f(t)) -@r(t)>, O<t,<T, i(t) =fWl -gM P>f(t)) -@r(t)), O<t<T,

(48) f(O) = N, r(O) = 0.

If the period T is short and g(F) = 1 - expC -F), we end up with the Ricker model in the tied strategy case and if k( p > = /3P with

N>- pMP ’ . 1

(49)


in the case of adjustable reproductive strategies. Gyllenberg et al. (1996) carried out a detailed analysis of the dynamical consequences of the adjustable strategy much in spirit of section 2 of the present paper, but they also extended their investigations to predator-prey models with optimal suppression of reproduction in the prey.

4. Relation to Commonly Used Discrete Models, Discrete single-species population models of the form

N’ =f(N), (50)

with a unimodal function f (e.g. the Ricker model) have in the literature been used to describe and explain both regular and chaotic population fluctuations. It is therefore of importance to investigate what kind of mechanisms might give rise to such models.

In this paper we have analyzed a class of models where reproduction is assumed to occur at discrete instants of time, whereas competition is assumed to be a continuous process. We have furthermore assumed a trade-off between reproductive effort and competitive ability: Individuals born in small clutches do better in competition than individuals born in large clutches. We have considered competition between types with different reproductive strategies. In particular, we have analyzed competition between fixed strategies, where individuals on average produce a fixed number of young, and adjustable strategies, where individuals are able to adjust the clutch size depending on the size of the total population.

We have shown that fixed reproductive strategies cannot lead to models of the form (50) with a non-monotone map f if one restricts to unstructured models with one trophic level. Adjustable strategies may very well lead to unimodal maps, but such strategies are not optimal and may be outcompeted.

Unstructured population models assume that all living individuals are equal at any moment. In particular, unstructured models do not distinguish between winners and losers in intraspecific competition. We have shown that a structured model that explicitly takes the dynamics of losers into account may give rise to a model of the form (50) with a non-monotone map f even in the case of a fixed reproductive strategy.

May and Oster (1976) collected the most commonly used discrete models in a table. We have reproduced the essential parts of this table in Table 1. The paper by May and Oster (1976) and in particular the table corresponding to Table 1 is widely referred to as a source of relevant discrete population models. However, one has to bear in mind that several of the authors referred to in Table 1 do not justify their models biologically. Moran (1950) did not use the exponential function form of the Ricker (1954) model, but only showed graphs of functions resembling the Ricker


Table 1. Commonly used discrete population models from May and Oster (1976)

Label Model Source

1 N’ = Nexp(r(1 -N/K))

2 N’ = N[l + r(1 -N/K)]

AN 3

N’ = 1+ exp[ -A(1 - N/B)1 4 N’ = hN(1 + uN)-~

AN 5 N’ =

1 + ( N/Bjb

6 N, = hN1-b, i

forN>C AN, for N < C

7 N’ = i

A,N, forN>K

A-N, for N<K

8 N’=[l/(a+bN)-al

Moran, 1950; Ricker, 1954; Macfadyen, 1963; Cook, 1965; May, 1974b Maynard Smith, 1968,1974a; May, 1973; Krebs, 1972; Li and Yorke, 1975; Chaundy and Phillips, 1936 Pennycuick et al., 1968; Usher, 1972; Beddington, 1974 Hassell, 1974; Hassell et al., 1976 Maynard Smith, 1974a; with b = 1, Leslie, 1957; Skellam, 1951; Utida, 1967

Varley et al., 1973 and references therein

Williamson, 1974 (with A+ > 1 and A_ < 1)

Utida, 1957

curve (4). Macfadyen (1963) does not use the exponential form either, but gives a reference to Ricker (1954). Chaundy and Phillips (1936) do not justify model 2 biologically at all. In their paper they considered the model N’ = aN2 + bN + c in a purely mathematical context, although a vague reference to genetic theory was given. A special case of the model 3 was considered more or less on an ad hoc basis by Pennycuick et al. (1968). This model was generalized by Usher (1972) into the form of model 3 and copied further by Beddington (1974). The model used by Utida (1967) is actually model 8, which was introduced by Utida (1957). We think that model 7, introduced in matrix form by Williamson (1974), is introduced more to generate oscillation than to explain anything of biological relevance. May and Oster (1976) also include a ninth model referring to an unpublished paper by Oster, Auslander and Guckenheimer, which we have not been able to trace. These models have been presented mostly as phenomenologi- cal models of population dynamics without considering the mechanism of competition, but the danger with this approach is that it may lead to biologically unrealistic models.

In the present paper we have attempted to provide explanations based on within-season competition for territory and either fixed or adjustable reproductive strategies for most of the models in Table 1. Model 1 corresponds to a situation where the season is short and the reproductive strategy is fixed. Model 2 is the pure scramble case (Hassell, 1976) for short seasons. For appropriate choices of the parameter values, models 4, 5, and 6 can be regarded as good approximations of our models (29) and (49) of an adjustable reproductive strategy.


This work was supported in part by The Academy of Finland, The Bank of Sweden Tercentenary Foundation, The Carl Trygger Foundation, The Niilo Helander Foundation and the NorFA Foundation. We are grateful to Horst Thieme for his constructive criticism on an earlier version of this paper and to Hans Metz for valuable comments which have considerably improved the paper. Mats Gyllenberg thanks Odo Diekmann for useful discussions in Oberwolfach.

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Received 7 March 1995 Revised version accepted 8 August 1996

Continuous versus discrete single species population models with adjustable reproductive strategies

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