On the mechanistic underpinning of discrete-time population models with Allee effect

ARTICLE IN PRESS

Theoretical Population Biology 72 (2007) 41–51 www.elsevier.com/locate/tpb

On the mechanistic underpinning of discrete-time population models with Allee effect Hanna T.M. Eskolaa,, Kalle Parvinena,b a

Department of Mathematics, FIN-20014 University of Turku, Finland b Turku Centre for Computer Science TUCS, Finland Received 18 August 2006 Available online 16 March 2007

Abstract The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261–269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resourceconsumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience. r 2007 Elsevier Inc. All rights reserved. Keywords: First-principles derivation; Allee effect; Beverton-Holt model; Hassell model; Logistic model; Ricker model; Skellam model

1. Introduction The term ‘Allee effect’, named after the work of (Allee, 1931, 1938; Allee et al., 1949), generally refers to a reduction in individual fitness at low population size or density. The effect takes place, when individuals benefit from the presence of conspecifics, and these benefits are lost as population densities decline. The mechanisms behind this might be, for example, predator dilution or saturation by abundant prey, cooperative predation, or facilitated mate-finding (Stephens et al., 1999). Strong Allee effects lead to threshold population densities, below which the population growth is negative and the population is likely to go extinct. For this reason, the Allee effect naturally has great importance in, for example, conservation biology, sustainable harvesting and pest control

Corresponding author.

E-mail addresses: hanna.eskola@utu.fi (H.T.M. Eskola), kalle.parvinen@utu.fi (K. Parvinen). 0040-5809/$ - see front matter r 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2007.03.004

(Stephens and Sutherland, 1999; Courchamp et al., 1999; Hopper and Roush, 1993; May, 1977). Allee effects are well represented in the scientific literature (see, e.g., Boukal and Berec, 2002 for a review). However, most of the models treated in the literature are phenomenological models, in which the Allee effect is merely incorporated in the model by the means of a term with convenient mathematical properties, leading to negative growth rates at low population densities (see, e.g., Asmussen, 1979; Hoppensteadt, 1982; Jacobs, 1984; Myers et al., 1995; Avile´s, 1999; Fowler and Ruxton, 2002; Barrowman et al., 2003; Liebhold and Bascompte, 2003; Schreiber, 2003). In deterministic continuous-time population models, there are some mechanistic models with Allee effect. For example, in the model presented originally in de Roos and Persson (2002) and extended in de Roos et al. (2003) and van Kooten et al. (2005), structured prey populations and stage-specific predation lead to Allee effects in the (top) predator population. The non-spatial model of a sexually reproducing species studied by Berec and Boukal (2004) also gives rise to the Allee effect, as do

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H.T.M. Eskola, K. Parvinen / Theoretical Population Biology 72 (2007) 41–51

the models given by Dennis (1989), in which different mating encounter processes have been taken into account. Further, Hopper and Roush (1993) have modelled dispersal explicitly with a two sex reaction–diffusion model, which exhibits the Allee effect. In addition, Thieme (2003, pp. 65–74) has given mechanistic models in continuous time with mate finding and predator saturation, in which the Allee effect occurs. However, in this paper, we are interested in discrete-time population models with Allee effects. We have also limited our study to deterministic population models, and for this reason will not give examples of stochastic population models. One way to derive discrete-time models mechanistically, also known as derivation from first principles, is to use sitebased models (Johansson and Sumpter, 2003; Bra¨nnstro¨m and Sumpter, 2005), but the models derived mostly do not exhibit an Allee effect. They can also exhibit an Allee effect, if one assumes that two individuals are needed for reproduction, but only few examples are published, including Eq. (30) in Johansson and Sumpter (2003) (Eq. (37) in this article). Another way is to incorporate a separate, continuous-time mating season into the reproductive period, and thus derive a probability of finding a mate and reproducing (Wells et al., 1990, 1998). We are not aware of any other mechanistic discrete-time population models with the Allee effect. Although the mechanisms giving rise to Allee effects are various, mating systems have a central status in them. The reason for this is that mating systems can affect the species’ vulnerability to Allee effects, but the Allee effects can also affect the species’ mating system (Stephens and Sutherland, 2000). Because of this, we have focused on the effects of added mate-finding in the resource-consumer model presented originally by Geritz and Kisdi (2004). There are also several studies concentrating explicitly on how mating systems affect the mating success, that is, the probability of actually finding a mate. For example, Dennis (1989) and McCarthy (1997) have derived encounter probabilities for different situations, and further, Bessa-Gomes et al. (2004) have studied the effects of sex ratio variation in detail. In this paper, we consider two mechanisms of mate finding, which can lead to the Allee effect, and add them to the resource-consumer model by Geritz and Kisdi (2004). These mechanisms are (i) a population consisting of two sexes, and (ii) pair formation. In order to obtain a higher variety of models, we have also added cannibalism to the original model. In Section 2, we first go briefly through the results given by Geritz and Kisdi (2004), and then present the extensions and the resulting discrete-time population models. In Section 3, we show analytically that neither the original model by Geritz and Kisdi (2004) nor the model extended with cannibalism gives rise to the Allee effect. In this section we also show that in the model extended with mate finding the Allee effect is always present, both with and without cannibalism; to use the terminology suggested by Stephens et al. (1999), we are interested in the occurrence of demographic Allee effects. In the concluding

Section 4 we give examples of the resulting discrete-time population models and compare them with the existing literature. 2. Mechanistic underpinnings 2.1. The model by Geritz and Kisdi (2004) The following resource-consumer model was presented by Geritz and Kisdi (2004) to give a unifying class of underpinnings of several well-known discrete-time population models. 2.1.1. Within-season dynamics Let Rn ðtÞ denote the resource population density within the season n at time t. Assume that in the absence of consumers, the resource population grows according to R_ n ðtÞ ¼ aRn ðtÞf ðRn ðtÞÞ.

(1)

The function f is assumed to be continuous and monotonically decreasing on ð0; 1Þ such that limR!0 Rf ðRÞ is finite and such that for some given positive K we have f ðRÞ40 if RoK and f ðRÞo0 if R4K. As a consequence, K is the unique asymptotically stable positive equilibrium density for the resource dynamics if no consumers are present. Consumers with population density xn ðtÞ harvest the resource with effort b according to the law of mass action, thus with a linear functional response. Consumers produce eggs at a per capita rate proportional to the food intake. The consumer population is thus assumed to reproduce asexually, since only one consumer individual is required for the egg production. The density of eggs accumulated since the beginning of the season is denoted by E n ðtÞ. With these assumptions we obtain the following differential equations for the within-year dynamics: 8 _ > < Rn ðtÞ ¼ aRn ðtÞf ðRn ðtÞÞ  Rn ðtÞbxn ðtÞ; E_ n ðtÞ ¼ gxn ðtÞbRn ðtÞ  dE n ðtÞ; (2) > :_ xn ðtÞ ¼ mxn ðtÞ:

2.1.2. Between-season dynamics All adults die in the end of the season, which is at time t ¼ 1. The population in the beginning of the next season will consist of those eggs which survive to the next season (for example the winter conditions can cause death) and hatch, which happens with probability s. Other eggs are assumed to be lost. The between-season survival probability for the resource is r. We have thus Rnþ1 ð0Þ ¼ rRn ð1Þ, E nþ1 ð0Þ ¼ 0, xnþ1 ð0Þ ¼ sE n ð1Þ.

(3)

ARTICLE IN PRESS H.T.M. Eskola, K. Parvinen / Theoretical Population Biology 72 (2007) 41–51

Geritz and Kisdi (2004) assumed that the within-year dynamics of the resource R is much faster than the withinyear dynamics of eggs E and consumers x. By using timescale separation, we observe that the resource densities have a unique asymptotically stable quasi-equilibrium, which is given by 8   > < f 1 bxn ðtÞ if 0pxn ðtÞpx ; a R^ n ðtÞ ¼ (4) > : 0 if xn ðtÞ4x ; where x ¼ ða=bÞ limR!0 f ðRÞ. We assume that the resource is at its quasi-equilibrium throughout the season, Rn ðtÞ ¼ R^ n ðtÞ. Then the differential equation (2) for the egg density becomes a first-order linear non-autonomous differential equation. The resulting discrete-time model (Geritz and Kisdi, 2004) is   Z 1 d ðdmÞt ^ xnþ1 ð0Þ ¼ xn ð0Þ sgbe Rn ðtÞe dt . (5) 0

We will later prove analytically (Lemma 1) that this model does not exhibit an Allee effect. In case m ¼ 0, the model (5) reduces to 8   > < lxn ð0Þf 1 bxn ð0Þ if 0pxn ð0Þpx ; a xnþ1 ð0Þ ¼ (6) > :0 if xn ð0Þ4x ; where l ¼ sgbð1  ed Þ=d. As shown by Geritz and Kisdi (2004), by choosing different resource growth models, Eq. (6) results in the discrete logistic model, the Ricker (1954) model, the Hassell (1975) model and the BevertonHolt model (Beverton and Holt, 1957) (See Table 2 with zðxn Þ ¼ xn ).

2.2. The extended model with cannibalism We have generalized the model by Geritz and Kisdi (2004) by adding cannibalism into the within-season dynamics. The consumers are assumed to harvest their own eggs with the rate k according to the law of mass action. Using the same notations as before, the within-year dynamics is now given by the following differential equations: 8 _ > < Rn ðtÞ ¼ aRn ðtÞf ðRn ðtÞÞ  Rn ðtÞbxn ðtÞ; _ E n ðtÞ ¼ gxn ðtÞbRn ðtÞ  dE n ðtÞ  kE n ðtÞxn ðtÞ; (7) > :_ xn ðtÞ ¼ mxn ðtÞ: This extension does not change the between-season dynamics and the resource quasi-equilibrium, which are thus given by (3) and (4), respectively. By using the same time-scale separation as in Geritz and Kisdi (2004), the differential equation for the egg density (7) becomes again a first-order linear non-autonomous differential equation,

43

and the resulting discrete-time model is  m xnþ1 ð0Þ ¼ xn ð0Þ sgbedþðk=mÞxn ð0Þe Z 

1

 mt R^ n ðtÞeðdmÞtðk=mÞxn ð0Þe dt .

ð8Þ

0

For consistency, note that the model (8) reduces to the original model (5) by Geritz and Kisdi (2004) if we set k ¼ 0. As is intuitively clear, the added cannibalism alone does not give rise to the Allee effect. This result is also proved analytically in Lemma 1. Also note that if we let m ! 0 equation (8) becomes xn ð0Þ xnþ1 ð0Þ ¼ sgbR^ n ð0Þ ð1  edkxn ð0Þ Þ d þ kxn ð0Þ   8 xn ð0Þ 1 b > > l xn ð0Þ f > > a < d þ kxn ð0Þ ¼ ð1  edkxn ð0Þ Þ if 0pxn ð0Þpx ; > > > > : 0 if xn ð0Þ4x ; ð9Þ where l ¼ sgb. If we furthermore assume that d ¼ 0 and the resource level R is independent of xn ð0Þ, and constant in time, equation (9) becomes the model of Skellam (1951) xnþ1 ð0Þ ¼ lð1  ekxn ð0Þ Þ,

(10)

where l ¼ ðsgb=kÞR. This result is consistent with the derivation of the Skellam model by Eskola and Geritz (2007). 2.3. Adult population consisting of two sexes In this section we present a modified model, which can result in discrete-time models with an Allee effect. The resource growth is assumed to be the same as in the model by Geritz and Kisdi (2004). The consumer population is assumed to consist of males M and females F. For simplicity, we assume that there is no adult mortality within the season ðm ¼ 0Þ. In a simple setting we assume that females produce eggs at a per capita rate proportional to the food intake and the male density. Therefore the positive term in the differential equation of the egg density becomes gF n ðtÞM n ðtÞbRn ðtÞ. However, this assumption is not very realistic for large male densities, since the egg production must eventually reach some (physiologically) maximal level, after which increasing male densities will have no effect. More generally, let this positive term be gF n ðtÞpðM n ðtÞÞbRn ðtÞ, where the function pðMÞ includes factors like the mating probability. For consistency, this function should also be mechanistically derived, and thus satisfy pð0Þ ¼ 0, p0 ð0Þ40 and pðMÞp1 for all M. For simplicity, we will concentrate on the case pðMÞ ¼ minðZM; 1Þ, where Z40 is a parameter related to the mating process. In addition to the model by Geritz and Kisdi (2004) we assume that both sexes consume the eggs (cannibalism) with the rate k. The differential equations for

ARTICLE IN PRESS H.T.M. Eskola, K. Parvinen / Theoretical Population Biology 72 (2007) 41–51

44

the within-year dynamics are thus 8 R_ n ðtÞ ¼ aRn ðtÞf ðRn ðtÞÞ > > > > > Rn ðtÞbðF n ðtÞ þ M n ðtÞÞ; > > |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} > > > xn ðtÞ > > > > < E_ n ðtÞ ¼ gF n ðtÞpðM n ðtÞÞbRn ðtÞ > dE n ðtÞ  kE n ðtÞðF n ðtÞ þ M n ðtÞÞ; > > |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} > > > xn ðtÞ > > > > > F_ n ðtÞ ¼ 0; > > > : _ M n ðtÞ ¼ 0:

Finally, we obtain 8 xn ð0Þpðð1  sÞxn ð0ÞÞ½1  eðdþkxn ð0ÞÞ  > > > l > > d þ kxn ð0Þ <   xnþ1 ð0Þ ¼ b 1 > xn ð0Þ f > > a > > : 0

(11)

if 0pxn ð0Þpx ; if xn ð0Þ4x ;

(15) where l ¼ sgbs and x ¼ ða=bÞlimR!0 f ðRÞ. If we assume pðMÞ ¼ minðZM; 1Þ, Eq. (15) becomes

Strictly speaking, a mechanistic derivation of such a situation would require a tri-molecular reaction. However, we could assume that before the egg production season there is a separate mating season, during which a female successfully mates with a male with probability pðMÞ (see Wells et al., 1990 for a mechanistic derivation of such a function). With this alternative assumption, we obtain otherwise Eq. (11), but we have pðM n ð0ÞÞ instead of pðM n ðtÞÞ in the differential equation describing egg production. But since we assume that _ n ðtÞ ¼ 0, we have M n ðtÞ ¼ M n ð0Þ for all t, 0ptp1. M This alternative assumption thus results in exactly the same model as described below. Another way to avoid a tri-molecular reaction is, for example, to include a requirement of a certain nutritional level on the reproducing females. Such a model is derived in the Appendix. Now, let xn ðtÞ ¼ F n ðtÞ þ M n ðtÞ denote the adult population size. The between-season dynamics is otherwise the same as in Geritz and Kisdi (2004), but now eggs become either females (fraction s) or males (fraction 1  s). We have thus Rnþ1 ð0Þ ¼ rRn ð1Þ,

8 minðZð1  sÞxn ð0Þ2 ; xn ð0ÞÞ½1  eðdþkxn ð0ÞÞ  > > > l > > d þ kxn ð0Þ <   xnþ1 ð0Þ ¼ b 1 > xn ð0Þ f > > a > > : 0

if 0pxn ð0Þpx ; if xn ð0Þ4x ;

(16) where l ¼ sgbs and x ¼ ða=bÞ limR!0 f ðRÞ. The situation without cannibalism can be obtained simply by setting k ¼ 0. In case pðMÞ ¼ minðZM; 1Þ the between-year dynamics becomes 8 l minðZð1  sÞxn ð0Þ2 ; xn ð0ÞÞ > >   > < 1 b xn ð0Þ if 0pxn ð0Þpx ; xnþ1 ð0Þ ¼ f > a > > :0 if xn ð0Þ4x ; (17) where l ¼ sgbsð1  ed Þ=d and x ¼ ða=bÞ limR!0 f ðRÞ. As will be shown in Lemma 2, the model (15) gives an Allee effect, if pð0Þ ¼ 0 and p0 ð0Þ40. The function pðMÞ ¼ minðZM; 1Þ satisfies these conditions, and therefore both models (16) and (17) give an Allee effect too. We shall return to it after we have looked at another related mechanism.

E nþ1 ð0Þ ¼ 0,

2.4. Pair formation

F nþ1 ð0Þ ¼ ssE n ð1Þ ¼ sxnþ1 ð0Þ,

In this section we present another model, which can also result in discrete-time models with an Allee effect. Again, the resource growth is assumed to be the same as in the model by Geritz and Kisdi (2004). Individual adults U search for mates with the rate c, and form a pair P. The species in question is thus assumed to be isogamous, and different male or female individuals cannot be distinguished; for the sake of completeness, we have also derived a corresponding two-sex model of a monogamous species in the Appendix. Pairs produce eggs at a per capita rate proportional to the food intake. The individual adults eat the eggs with the rate k. For simplicity, we assume that there is no adult mortality within the season. We also assume that there is no egg mortality except for the mortality caused by cannibalism; this is a technical assumption made to ensure that the related differential equations can be solved analytically. The differential

M nþ1 ð0Þ ¼ ð1  sÞsE n ð1Þ ¼ ð1  sÞxnþ1 ð0Þ.

(12)

Assuming fast resource dynamics results in a stable quasiequilibrium given in Eq. (4), where xn ðtÞ ¼ F n ðtÞ þ M n ðtÞ ¼ F n ð0Þ þ M n ð0Þ ¼ xn ð0Þ. Therefore we obtain for the egg densities E_ n ðtÞ ¼ C  ðd þ kxn ð0ÞÞEðtÞ

(13)

with C ¼ gF n ð0ÞpðM n ð0ÞÞbR^ n ð0Þ ¼ gsxn ð0Þpðð1  sÞxn ð0ÞÞbR^ n ð0Þ. The solution of (13) with E n ð0Þ ¼ 0 is E n ðtÞ ¼ C

ð1  etðdþkxn ð0ÞÞ Þ . ðd þ kxn ð0ÞÞ

(14)

ARTICLE IN PRESS H.T.M. Eskola, K. Parvinen / Theoretical Population Biology 72 (2007) 41–51

equations for the within-year dynamics are thus 8 R_ n ðtÞ ¼ aRn ðtÞf ðRn ðtÞÞ  Rn ðtÞbðU n ðtÞ þ 2Pn ðtÞÞ; > > > > < E_ n ðtÞ ¼ gPn ðtÞbRn ðtÞ  kU n ðtÞE n ðtÞ; U_ n ðtÞ ¼ cU n ðtÞ2 ; > > > > : P_ n ðtÞ ¼ 1 cU n ðtÞ2 :

(18)

2

The between-season dynamics is essentially the same as in Geritz and Kisdi (2004) Rnþ1 ð0Þ ¼ rRn ð1Þ, E nþ1 ð0Þ ¼ 0, U nþ1 ð0Þ ¼ sE n ð1Þ, Pnþ1 ð0Þ ¼ 0.

ð19Þ

Assuming fast resource dynamics results in a stable quasiequilibrium given in Eq. (4), where xn ðtÞ ¼ U n ðtÞ þ 2Pn ðtÞ ¼ U n ð0Þ ¼ xn ð0Þ. Solving the differential equations for U n ðtÞ and Pn ðtÞ results in U n ðtÞ ¼

xn ð0Þ 1 þ cxn ð0Þt

Pn ðtÞ ¼

and

2 1 2 cxn ð0Þ t

1 þ cxn ð0Þt

,

(20)

k=c



.

(21)

For the between-season dynamics we finally obtain 8 > l½kxn ð0Þ  1 þ ð1 þ cxn ð0ÞÞk=c  > >   < b xnþ1 ð0Þ ¼ f 1 xn ð0Þ > a > > : 0

if 0pxn ð0Þpx ; if xn ð0Þ4x ;

(22) where l ¼ sgbc=2kðc þ kÞ and x ¼ ða=bÞ limR!0 f ðRÞ. The situation without cannibalism, i.e. k ¼ 0, must be solved separately. The solutions for U n ðtÞ and Pn ðtÞ are as in (20), but now the differential equation for the egg densities is E_ n ðtÞ ¼ gPn ðtÞbR^ n ð0Þ. The solution for this is

(24) where l ¼ sgb=2 and x ¼ ða=bÞ limR!0 f ðRÞ. For consistency, note that Eq. (24) can be obtained from Eq. (22) by taking the limit k ! 0. In Lemma 2 we show analytically that also models (22) and (24) give an Allee effect. 3. Presence and absence of the Allee effect in the resulting discrete-time models In the previous section we have obtained several discrete-time population models, which can in general be written as

cxn ð0Þt  lnð1 þ cxn ð0ÞtÞ , 2c

(23)

20

2

15

1.5

10

1

(25)

Obviously an extinct population will remain extinct, which means that hð0Þ ¼ 0. The Allee effect means increasing per capita growth at low densities, which means that g0 ð0Þ40 or equivalently h00 ð0Þ40 (see also Fig. 1). The following calculation shows that the discrete-time model (5) derived by Geritz and Kisdi (2004) and the model (8) with added cannibalism have no Allee effect. Lemma 1. (1) In the discrete-time model (5) by Geritz and Kisdi (2004) we have h00 ð0Þo0, thus there is no Allee effect. (2) In the discrete-time model (8) with cannibalism we also have h00 ð0Þo0, thus there is no Allee effect. Proof. As x_ n ðtÞ ¼ mxn ðtÞ, the function xn ðtÞ is an increasing function of xn ð0Þ. Furthermore, as the function f is assumed to be monotonically decreasing, the inverse function f 1 is monotonically decreasing as well. Combining these two results, we find that R^ n ðtÞ is a decreasing function of xn ð0Þ. Finally, by differentiating h, in case (1)

No Allee effect: g (x)<0 for all x

(0

0.5

Allee effect:

h"

5

)<

0

E n ðtÞ ¼ gbR^ n ð0Þ

which results in  8  1 > > l xn ð0Þ  lnð1 þ cxn ð0ÞÞ > > c > <   b xnþ1 ð0Þ ¼ 1 f xn ð0Þ if 0pxn ð0Þpx ; > > > a > > : 0 if xn ð0Þ4x ;

xnþ1 ð0Þ ¼ xn ð0Þgðxn ð0ÞÞ ¼ hðxn ð0ÞÞ.

and for the egg densities we obtain the solution c½kxn ð0Þt  1 þ ð1 þ cxn ð0ÞtÞ E n ðtÞ ¼ gbR^ n ð0Þ 2kðc þ kÞ

45

h" (0) > 0

0 0

5

g (x) > 0 for small x

0 10 x

15

20

0

5

10 x

15

20

Fig. 1. Illustration of discrete-time population models xnþ1 ¼ xn gðxn Þ ¼ hðxn Þ with (thick curve) and without (thin curve) an Allee effect. (a) Population size hðxn Þ; (b) per capita gðxn Þ ¼ hðxn Þ=xn .

ARTICLE IN PRESS H.T.M. Eskola, K. Parvinen / Theoretical Population Biology 72 (2007) 41–51

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we obtain h00 ð0Þ ¼ 2sgbed

Z 1 0

 q R^ n ðtÞ eðdmÞt dto0. qxn ð0Þ

In case (2) we obtain  Z 1  q 00 d ^ Rn ðtÞ h ð0Þ ¼ 2sgbe qxn ð0Þ 0  ðdmÞt ðdmÞt k mt m ðe  e Þ dto0,  Ke e m

(26)

ð27Þ

because emt Xem for all t 2 ½0; 1. & In the five models with mate finding derived in this paper, the Allee effect is present: Lemma 2. In the discrete-time models (16), (17), (22) and (24) we have h0 ð0Þ ¼ 0 and h00 ð0Þ40, thus the Allee effect is present. In the discrete-time model (15) we have h0 ð0Þ ¼ 0 and 00 h ð0Þ40 if pð0Þ ¼ 0 and p0 ð0Þ40. Proof. All these five models are of form   1 b xn ð0Þ ¼ hðxn ð0ÞÞ, xnþ1 ð0Þ ¼ lzðxn ð0ÞÞ f a

(28)

when 0pxn ð0Þpx . It is easy to see that zð0Þ ¼ 0 and z0 ð0Þ ¼ 0 for all the models. Therefore, the first derivative h0 ð0Þ ¼ 0, and the second derivative h00 ð0Þ ¼ lz00 ð0Þf 1 ð0Þ ¼ lz00 ð0ÞK. Let us first look at the general model (15) of the population consisting of two sexes. There we have h00 ð0Þ ¼ 2lKp0 ð0Þ40,

Table 1 Mating systems and functions zðxn Þ Mating system

Function zðxn Þ

Parameter l

Asexual (Geritz and Kisdi, 2004) Asexual with cannibalism

z1 ðxn Þ ¼ xn

sgb

Two-sex model Two-sex model with cannibalism Pair formation

xn ð1  edkxn Þ z2 ðxn Þ ¼ dþkx n

z3 ðxn Þ ¼ minðZð1  z4 ðxn Þ ¼

1ed d

sgb

sÞx2n ; xn Þ

minðZð1sÞx2n ;xn Þ½1eðdþkxn Þ  dþkxn

z5 ðxn Þ ¼ xn  1c lnð1 þ cxn Þ

Pair formation with z6 ðxn Þ ¼ kxn  1 þ ð1 þ cxn Þ cannibalism Generalized two-sex model z ðx Þ ¼ xn pðð1sÞxn Þ½1eðdþkxn Þ   n dþkxn with cannibalism (requires a mechanism for pðMÞ)

k=c

sgbs sgbs

1ed d

sgb 2 sgbc 2kðcþkÞ

sgbs

(29)

because p0 ð0Þ is assumed to be positive, and l ¼ sgbsð1  sÞ ð1  ed Þ=d40. In the special cases (16) and (17) we have pðMÞ ¼ minðZM; 1Þ. Therefore p0 ð0Þ ¼ Z40, and h00 ð0Þ40 as well. Note that cannibalism does not affect the quantity h00 ð0Þ. Also in the case of a system with pair formation (i.e., for models (22) and (24)), cannibalism does not affect the quantity h00 ð0Þ, which is h00 ð0Þ ¼ lcK40,

where x ¼ ða=bÞlimR!0 f ðRÞ. The function zðxn Þ and the parameter l depend on the choice of the mating system and individual-level parameters, as summarized in Table 1. The function f 1 ðyÞ depends on the resource growth model. Various examples are given in Table 2. Functions z1 and z2 (see Table 1) resulting in models without the Allee effect are plotted in Fig. 2a. The function z1 is linear and the function z2 is linear near the origin, and lz2 ðxÞ ! l=d as x ! 1, thus it has a horizontal asymptote. The functions z3 and z4 derived from the two-sex model are illustrated in Fig. 2b. For x41=Zð1  sÞ their shapes correspond to functions z1 and z2 , respectively. They have thus the same asymptotes. The functions z5 and z6 derived from the model with pair formation are illustrated in Fig. 2c.

(30)

Table 2 Various continuous-time models for the resource and the resulting discrete-time models for the consumer Resource growth R_ n ðtÞ ¼

Resulting model xnþ1 ¼

Continuous logistic   Rn ðtÞ aRn ðtÞ 1  K

Discrete logistic-type 8   > < lKzðx Þ 1  bx if 0pxn px n n a > :0 if xn 4x

Gompertz equation ðK41Þ   ln Rn ðtÞ aRn ðtÞ 1  ln K

Ricker-type

Von Bertalanffy ð0oyo1Þ

Hassell-type

aRyn ðtÞ  aK y1 Rn ðtÞ

lKzðxn Þ ; ð1 þ bxn Þc

Constant influx with decay

Beverton-Holt-type

where l is now given by l ¼ sgb=240. & 4. Conclusion 4.1. Summary of the derived models To summarize, we have presented several different modifications of the model by Geritz and Kisdi (2004), which give rise to discrete-time population models of the form 8   > < lzðx Þf 1 b x if 0pxn px ; n n a xnþ1 ð0Þ ¼ (31) > :0 if xn 4x ;

a

aRn ðtÞ K

lKzðxn Þebxn ;

lKzðxn Þ ; 1 þ bxn

b b ¼ ln K a

b 1 b ¼ K 1y ; c ¼ a 1y

b b¼ K a

(32)

(33)

(34)

(35)

ARTICLE IN PRESS H.T.M. Eskola, K. Parvinen / Theoretical Population Biology 72 (2007) 41–51

z1 and z2

z3 and z4 5

λz

3

(x ) λz

1

6

4

λz2 (x)

3

λz4 (x)

4

4 2

z5 and z6

(x )

8

8 6

47

2

2

λz 5

(x)

(x) λz 6

1 0

0

2

4

6

8

10

12

14

0

2

4

6

x

8

10

12

14

0

x

2

4

6

8

10

12

14

x

Fig. 2. Illustration of functions lzðxn Þ in different mating systems. Parameters: d ¼ 0:2, k ¼ 0:3, Z ¼ 0:3, s ¼ 0:5, c ¼ 0:1. In panel (b) we observe that for x41=Zð1  sÞ  6:67, the shapes of functions z3 and z4 (solid curves) correspond to those of z1 and z2 (dashed curves). The dashed lines in panels (a) and (c) describe asymptotes.

Function z6 has a linear asymptote. The function z from the generalized two-sex model is not illustrated. According to Lemma 2 the functions z3 , z4 , z5 , z6 , and z are quadratic near the origin, and thus result in an Allee effect. 4.2. Comparison with previous studies Allee effects are well represented in the scientific literature (cf. Boukal and Berec, 2002). However, only few articles provide a first-principles derivation of the model used (Wells et al., 1998; Johansson and Sumpter, 2003; Bra¨nnstro¨m and Sumpter, 2005). Quite commonly in the literature the Allee effect is merely incorporated in the model by the means of a term with convenient mathematical properties. These models are thus phenomenological models lacking a mechanistic underpinning (see, e.g., Asmussen, 1979; Hoppensteadt, 1982; Jacobs, 1984; Myers et al., 1995; Avile´s, 1999; Fowler and Ruxton, 2002; Barrowman et al., 2003; Liebhold and Bascompte, 2003; Schreiber, 2003). We will first discuss other mechanistic models in more detail, and then study some phenomenological models. The phenomenological models discussed here are not supposed to represent an exhaustive list of models in the literature; we merely want to give some examples of what our modelling framework can and what it cannot explain. 4.2.1. Separate mating season within the reproductive period Wells et al. (1998) give a mechanistic derivation for a discrete-time population model by assuming a separate continuous-time mating season within the reproductive period. The original context was to model the importance of overwinter aggregation for reproductive success of monarch butterflies (Wells et al., 1990), and the probability of not finding a mate by time t within a reproductive period was derived by using a pair of differential equations to describe the mating process. The model derived by Wells et al. (1998) is of the form xnþ1 ¼

lx2n þ oxn , axn þ 1

(36)

where o is the probability of an individual surviving to the next reproductive period. Our two-sex model (17) results in the same model on the domain xn 2 ½0; 1=ðð1  sÞZÞ when there is constant influx with decay in the resource (Eq. 35), mating probability function is chosen to be pðMÞ ¼ minðZM; 1Þ, which results in zðxn Þ ¼ minðZð1  sÞ x2n ; xn Þ, and a fraction o of the adult consumer individuals is allowed to survive to the next season. The derivation of (36) by Wells et al. (1998) can be implemented in our framework in the following way: Our two-sex model (15) becomes xnþ1 ¼ lxn pðð1  sÞxn Þ in the special case that there is no cannibalism (k ¼ 0), and that the resource level R is independent of xn , and constant in time. The choice of the function p according to Wells et al. (1990) results in (36), when the fraction o of the adult consumer individuals is again allowed to survive to the next season. 4.2.2. Site-based models Another way to derive discrete-time models from first principles is to use site-based models (Johansson and Sumpter, 2003; Bra¨nnstro¨m and Sumpter, 2005). The Allee effect does not occur in most of the models derived this way, but the models can also exhibit Allee effects if one assumes that two individuals are needed for reproduction. However, only few examples are published. One of these is Eq. (30) in Johansson and Sumpter (2003): xnþ1 ¼ l½1  ð1 þ xn Þexn .

(37)

We do not obtain this equation within our setting. Another example is the Allee-effect model mentioned in the discussion of Bra¨nnstro¨m and Sumpter (2005), namely xnþ1 ¼ l

x2n , að1 þ xn =aÞ2

(38)

although they did not explicitly write it down in the article. In our setting, the two-sex model given by Eq. (17) with the Von Bertalanffy growth in the resource gives also Eq. (38) in the domain xn 2 ½0; 1=ðð1  sÞZÞ. This can easily be seen in Eq. (34) by choosing y ¼ 12 and zðxn Þ ¼ min

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48

ðZð1  sÞx2n ; xn Þ. It is interesting to obtain the same form of model from two different mechanisms.

with other exponents than 2: xnþ1 ¼ l

4.2.3. Examples of non-mechanistic models which can be derived within our framework There are various published models without a mechanistic underpinning, which can be derived within our framework. For example the following Ricker-type model with an Allee effect: cxn xnþ1 ¼ lx1þg n e

(39)

has been studied by Asmussen (1979) with g ¼ 1 and by Avile´s (1999) with 0ogp1. Like in the cases of Eqs. (40) and (42), the two-sex model without cannibalism and with Gompertz growth in the resource given by Eq. (33) with zðxn Þ ¼ minðZð1  sÞx2n ; xn Þ provides a mechanistic derivation for the Eq. (39) in the domain xn 2 ½0; 1=ðð1  sÞZÞ in the case g ¼ 1. The similar modification of the discrete logistic model  xn  xnþ1 ¼ lx1þg 1  (40) n K has also been studied by Avile´s (1999). Similarly as above, our two-sex model without cannibalism and with logistic growth in the resource given by Eq. (32) with zðxn Þ ¼ minðZð1  sÞx2n ; xn Þ provides a mechanistic derivation in the domain xn 2 ½0; 1=ðð1  sÞZÞ for the case g ¼ 1. Note that all mechanistic models with an Allee effect derived in this article are quadratic at the origin. This property holds for models (43) and (44) only in the case g ¼ 1. Fowler and Ruxton (2002) studied the Hassell-type model xnþ1 ¼ lxn

1  Aebxn =g . ð1 þ bxn Þc

(41)

In our framework, the function z in the two-sex model with cannibalism and d ¼ 0 is z4 ðxn Þ ¼ ðZð1  sÞ=kÞxn ½1  ekxn  in the domain xn 2 ½0; 1=ðð1  sÞZÞ. By choosing the Von Bertalanffy resource growth (34) we obtain (41) with A ¼ 1. Again, this is the only case where (41) is quadratic at the origin. 4.2.4. Examples of non-mechanistic models which cannot be derived within our framework There are naturally several phenomenological models, which cannot be explained using the setting in this article for one reason or another. For example, Avile´s (1999) studied the model h  xn i xnþ1 ¼ xn xgn þ r 1  ; 0ogo1. (42) K This is a modification of the discrete logistic model, into which the term xgn has been added. This term cannot be interpreted as an adult survival probability as in Eq. (36). Therefore it cannot be derived using our framework. Another example is the following model presented by Hoppensteadt (1982), which was also studied by Jacobs (1984). Myers et al. (1995) studied a generalized version

x2n . A þ x2n

(43)

The Eq. (43) is rather similar to Eq. (38) which can be explained using site-based models and our framework. However, in order to explain (43) using our framework, we need to have zðxn Þ ¼ x2n and resource growth according to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R_ n ðtÞ ¼ aRn ðtÞ  A ¼ a Rn ðtÞð1  ARn ðtÞÞ (44) Rn ðtÞ for 0oRn ðtÞo1=A. Technically, the model (43) can thus be derived. However, in order to be consistent, the resource growth model (44) should have a mechanistic underpinning as well. We are not aware of any mechanisms that would result in (44), and thus the model (43) cannot be explained using our framework. 4.2.5. Conclusion In this article, we have presented a generator of relatively realistic discrete-time Allee effect models. Since the parameters in mechanistic models have a direct connection to the individual behavior, the use of these models might prove helpful in applications to real, biological systems. As already stated in Eskola and Geritz (2007), different derivations for the models do not take anything away from each other. Instead, knowledge of the different mechanisms might make the choice of an appropriate model easier and more consistent based on the biological background of the system at hand. The aim of this paper is not to be revolutionary in relation to the previously published studies with Allee effects, but instead to offer a complementary approach. As the already published mechanistic discrete-time population models have often been for special cases (Wells et al., 1998) or using a totally different approach (site-based models, e.g., Johansson and Sumpter, 2003; Bra¨nnstro¨m and Sumpter, 2005), we offer a single, general framework in which several population models can be derived. We use discrete-time population models, because they might be more representative of annual species, whereas continuous models can be more adequate in other scenarios. The pair-formation model is derived for isogamous species, but it might also be used to approximate the population density of a species with two separate sexes, if the population sex ratio s is 12 (see Eq. (B.6) in the Appendix). The important criterion in this case is that the species behaves monogamously. On the other hand, the two-sex model represents a promiscuous situation, with the egg production increasing (boundedly) with increasing male density. In this case, one can use mating probabilities derived in, for example, Dennis (1989) and McCarthy (1997), as the function pðMÞ. In both these cases, we have given explicit population dynamics already including the Allee effect. In theory, researchers working with real data could use these models in two ways. If population densities

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49

have been measured, then the models could be used to estimate the model parameters by data fitting. On the other hand, if the parameters related to individual behavior (such as the cannibalism rate, resource consumption, etc.) are known, then the models could be used to estimate and predict population densities. The details depend on the specific field of application.

and the between-year dynamics is otherwise the same as in our original two-sex model given in Eq. (12), but with the addition of F nþ1 ð0Þ ¼ 0. As before, assuming fast resource dynamics results in a stable quasi-equilibrium given in Eq. (4) with xn ðtÞ ¼ xn ð0Þ. First, we can see that F n ðtÞ ¼ sxn ð0Þ  F n ðtÞ, and if we denote   b Cðxn ð0ÞÞ ¼ bf 1 xn ð0Þ þ pðð1  sÞxn ð0ÞÞ, (A.2) a

Acknowledgments

we can solve for the satiated females   b bsxn ð0Þf 1 xn ð0Þ a ð1  eCðxn ð0ÞÞt Þ. F n ðtÞ ¼ Cðxn ð0ÞÞ

The authors wish to thank Stefan Geritz (especially for ideas and comments concerning Section 2.3), David Boukal and A˚ke Bra¨nnstro¨m for discussions. The authors also wish to thank the Finnish Graduate School in Computational Biology, Bioinformatics, and Biometry (ComBi) and the Academy of Finland for financial support.

The corresponding between-year dynamics

  8 1 b > > x ð0ÞÞx ð0Þf ð0Þ bgspðð1  sÞx n n n > > a > < ðkxn ð0ÞþdÞ ðCðxn ð0ÞÞð1  e Þ  ðkxn ð0Þ þ dÞð1  eCðxn ð0ÞÞ ÞÞ xnþ1 ð0Þ ¼ >  > > Cðxn ð0ÞÞðCðxn ð0ÞÞ  kxn ð0Þ  dÞðkxn ð0Þ þ dÞ > > : 0

Appendix A. The two-sex model with a requirement on the initial nutritional level In this Appendix we give an alternative mechanism for the two-sex model, in which we do not use a tri-molecular reaction. Assume that the females, F, cannot produce eggs until they have reached a satiated stage, F  , which is achieved by the means of resource consumption. Reproduction then happens, as satiated females encounter males (with the function pðMÞ as in the original two-sex model), and females also return to the ‘hungry’ state F after reproducing. The initial nutritional level can be interpreted as extra resources needed for reproduction, and thus the satiated females are also assumed to consume the resource. In addition, all adults are assumed to cannibalize the eggs. The within-year dynamics is now given by the following differential equations: 8 R_ n ðtÞ ¼ aRn ðtÞf ðRn ðtÞÞ > > > > > Rn ðtÞbðF n ðtÞ þ F n ðtÞ þ M n ðtÞÞ; > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > > xn ðtÞ > > > >  _ > > < E n ðtÞ ¼ gF n ðtÞpðM n ðtÞÞ dE n ðtÞ  kE n ðtÞðF n ðtÞ þ F n ðtÞ þ M n ðtÞÞ; (A.1) |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > xn ðtÞ > > > >  > _ > ðtÞ ¼ bR ðtÞF ðtÞ þ F ðtÞpðM F n n n n ðtÞÞ; > n > > >   > F_ n ðtÞ ¼ bRn ðtÞF n ðtÞ  F n ðtÞpðM n ðtÞÞ; > > > : _ M n ðtÞ ¼ 0

(A.3)

if 0pxn ð0Þpx ;

(A.4)

if xn ð0Þ4x ;

where x ¼ ða=bÞ limR!0 f ðRÞ, results in the Allee effect, as expected. This can be seen in the following way. If we denote xnþ1 ð0Þ ¼ hðxn ð0ÞÞ, the second derivative of hðxn ð0ÞÞ with respect to xn ð0Þ is given by

 2sgeKb ð1  sÞs eKbd ðed  1ÞKb  dðeKb  1Þ p0 ð0Þ . h ð0Þ ¼ dðKb  dÞ 00

(A.5) Because eKbd ðed  1ÞKb  dðeKb  1Þ and Kb  d always have the same sign, then as p0 ð0Þ40, the numerator and the denominator in Eq. (A.5) have always the same sign. Thus, h00 ð0Þ40, and the Allee effect is present in the model (see Section 3). However, this model seems a bit too complicated for practical purposes, and it is given here mainly for the sake of completeness.

Appendix B. Pair formation with two sexes In this Appendix we give a model for a monogamous population with two sexes. Assume thus that the population consists of males M and females F, which form pairs with the rate c. In order to be able to solve explicitly the system, we have not included any egg mortality in the model. The within-year dynamics of the population is now given by the following system of differential

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50

equations: 8 _ Rn ðtÞ ¼ aRn ðtÞf ðRn ðtÞÞ  Rn ðtÞbðF n ðtÞ þ M n ðtÞ þ 2Pn ðtÞÞ; > > > > _ > > < E n ðtÞ ¼ gPn ðtÞbRn ðtÞ; F_ n ðtÞ ¼ cF n ðtÞM n ðtÞ; > > _ n ðtÞ ¼ cF n ðtÞM n ðtÞ; > M > > > : _ Pn ðtÞ ¼ cF n ðtÞM n ðtÞ: (B.1) The between-year dynamics is otherwise the same as in our original two-sex model given in Eq. (12), but with the addition of Pnþ1 ð0Þ ¼ 0. First, one can solve that M n ðtÞ ¼ F n ðtÞ þ ð1  2sÞxn ð0Þ, and thus get the solution sð1  2sÞxn ð0Þ ,  sÞ  s ð1  sÞð1  2sÞxn ð0Þ M n ðtÞ ¼ . ð1  sÞ  ecxn ð0Þtð2s1Þ s

F n ðtÞ ¼

ecxn ð0Þtð12sÞ ð1

ðB:2Þ

From this, one can also solve for the pair density Pn ðtÞ ¼

sð1  sÞxn ð0Þð1  ecxn ð0Þtð2s1Þ Þ ð1  sÞ  secxn ð0Þtð2s1Þ

(B.3)

and the corresponding between-season dynamics becomes 8 l½cð1  sÞxn ð0Þ > > >   > > s  ð1  sÞecxn ð0Þð12sÞ > > >  ln < 2s  1   xnþ1 ð0Þ ¼ > 1 b > > x f ð0Þ if 0pxn ð0Þpx ; n > > a > > > :0 if xn ð0Þ4x ; (B.4) where l ¼ sgb=c and x ¼ ða=bÞ limR!0 f ðRÞ. If we denote xnþ1 ¼ hðxn ð0ÞÞ and take the second derivative with respect to xn ð0Þ, we get h00 ð0Þ ¼ cð1  sÞsbgK40,

(B.5)

when 0oso1. Thus, this model also exhibits the Allee effect (see Section 3). Note that if we let the sex ratio s ! 12, the betweenseason dynamics becomes 8    1 1 > > ln 1 þ cx l x ð0Þ  ð0Þ > n n > > c 2 <   xnþ1 ð0Þ ¼ b > if 0pxn ð0Þpx ; f 1 xn ð0Þ > > a > > : 0 if xn ð0Þ4x ;

(B.6) where l ¼ sgb=2 and x ¼ ða=bÞ limR!0 f ðRÞ. This is almost the same as Eq. (24) for an isogamous population, except for the one factor 12 in the logarithmic function. This factor results from the fact that in an isogamous population all encounters between individuals may lead to pair formation, whereas in a population with two sexes only half of the encounters are with a possible mate.

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