Документ взят из кэша поисковой машины. Адрес оригинального документа : http://ecology.genebee.msu.ru/3_SOTR/CV_Terekhin_publ/2008_R0r_EcolMod.pdf Дата изменения: Mon Mar 16 11:14:34 2009 Дата индексирования: Mon Oct 1 21:21:33 2012 Кодировка:

This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author's institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright


Author's personal copy
ecological modelling 2 1 2 (2008) 5­9

available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/ecolmodel

The impact of different density stresses on the dynamics of two competitive populations
Anatoly T. Teriokhin , Elena V. Budilova
Department of Biology, Moscow Lomonosov State University, Moscow 119992, Russia

article
Article history:

info

abstract
We compare the asymptotic dynamics of two competitive populations described by a system of two differential equations under different density stresses in order to clarify whether some parameters of the system or a function of them can be interpreted as a fitness criteria which a population should optimise to win the competition. Four types of stress are considered. First, it is an age-independent thinning of individuals that prevents that the common density of two competing populations would be greater than some limit value K. The second type of stress is a thinning of youngs and the third one is a thinning of adults. These three stresses come into play just when the limit is reached. The forth stress, like the first one, does not depend on age but, unlike it, begins to take action gradually and increases gradually as density approaches K. We show that in the first case the population with a greater Malthusian coefficient r will ultimately supplant the other one. In contrast, in the second and third cases, the greater value of r does not ensure winning but instead that population whose lifetime reproductive success, R0 , is higher will supplant the other. And, at last, in the fourth case neither greater r nor R0 guarantee the win. Instead, that population whose environmental carrying capacity is greater than the capacity of the other population will supplant the other one. © 2007 Elsevier B.V. All rights reserved.

Published on line 26 November 2007 Keywords: Competition Invasion Fitness r-Selection R0 -selection K-selection

1.

Introduction

The question we try to answer here can be formulated as follows. Let there be two competing populations X and Y composed of individuals with phenotypes px and py , correspondingly. Does there exist a phenotype-dependent function or a functional, called fitness criterion, such that the inequality F(py )> F(px ) implies that the population Y will inevitably supplant the population X?Ifso, F may be interpreted as a criterion of evolutionary selection. We will see that, indeed, in some model situations such criteria exist but they can be different for different models. The key role is played by the mechanism of braking or limiting the growth of population density. We consider several types of density-dependent environmental

stress: age-independent thinning, thinning of youngs, thinning of adults, and a model of a steadily increased density stress. It is convenient, though not necessary, to imagine X as a large resident population and Y as a small population of invaders. The question then is what performances of the invaders can ensure that they supplant the residents. Or, alternatively, we may consider X as a large initial population and Y as a small population of mutants. If a fitness criterion exists and its value is greater for Y than for X then Y will supplant X and become resident. But another mutant Z may appear with still greater fitness and then Z will supplant Y, and so on. So, with such successive replacements of genotypes, the fitness criterion will grow tending to its maximum.

Corresponding author. E-mail address: terekhin a@mail.ru (A.T. Teriokhin). 0304-3800/$ ­ see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2007.10.013




Author's personal copy
6
ecological modelling 2 1 2 (2008) 5­9

This is what is called evolutionary optimization (e.g. Teriokhin and Budilova, 2001; Teriokhin et al., 2003). Among the firsts, Mylius and Diekmann (1995) drew attention to the importance of taking into account the nature of density stress when considering. They analyzed the situations of different density stresses using the Maynard Smith approach based on the game theory and the notion of evolutionary stable strategy (Maynard Smith, 1982). Here we use a more straightforward and, as we believe, intuitively more perceptible approach, based on the study of the population asymptotic dynamics in different two-populations competition models.

2.1.

Age-independent thinning

Rewrite system (1) as follows xt yt
+dt +dt

= xt + rx xt dt = yt + ry yt dt

(2)

and represent the mechanism of proportional reducing the density of youngs and adults as multiplying the right-hand sides of both equations of the system by some coefficient (intuitively this process can be imagined as thinning the total population by a factor of 1/ applied with the same intensity to the youngs and to the adults of the both populations X and Y): xt
+dt +dt

2.

Models of stopped exponential growth
yt

= (xt + rx xt dt) = (yt + ry yt dt)

(3)

The model of simple exponential growth is based on the assumption that the rate of growth of population density is proportional to the density (we assume that the common ecotope of two competitive population is fixed so that equations for density and size are equivalent):

The condition that the equality xt + yt = K attained at the moment t cannot be upset at the following moment t +dt implies that the following equality: xt
+dt

dx = rx xt dy = r y yt
dt dt

+ yt

+dt

= xt + yt

(4)

(1)

should be true as soon as the limit K is attained. Replacing xt+dt and yt+dt in (4) with their expressions from (3) we obtain the following equation for calculating : (xt + rx xt dt + yt + ry yt dt) = xt + yt from which we get: = xt + yt xt + rx xt dt + yt + ry yt dt (5)

where xt and yt are the densities of X and Y at time t, and rx and ry are the so-called intrinsic rates of population growth which can be expressed as differences rx =bx - dx and ry =by - dy , of birth rates bx and by and death rates dx and dy (e.g. May, 2001). Evidently, the model of unlimited exponential growth (1) is unrealistic for very big xt and yt because of limited environmental resources. Hence it should be modified to take into account such limitations. This can be made in several ways. Assume that the nature of the environment is such that the sum of densities of X and Y cannot exceed some constant K, the carrying capacity of the environment. We will suppose that while the condition xt + yt < K is true, the densities xt and yt grow exponentially in accordance with Eq. (1). But just at the moment when the environment is saturated, i.e. the equality xt + yt = K is attained, some mechanism comes into force which does not allow for the total density to exceed K. At this moment the populations X and Y begin to compete because the limitation is imposed on their sum so that an increase of one population prevents automatically an increase of another one. Our purpose consists in comparing the dynamics of two populations as dependent on the parameters rx and ry representing their phenotypes. Especially, we are interested in finding conditions under which Y will supplant X. The mechanism of limiting the sum of densities of X and Y can be realized at least by three ways. First, it is possible to reduce in the same proportion both the density of youngs and the density of adults. Second, only the density of youngs can be reduced. And third, the required limitation can be ensured by reducing only the density of adults.

Substituting (5) into (3) and subtracting xt and yt from the both sides of the first and second equations in (3), we obtain:

xt y t

+dt

= =

(xt + yt )(xt + rx xt dt) - xt xt + rx xt dt + yt + ry yt dt (xt + yt )(yt + ry yt dt) - yt xt + rx xt dt + yt + ry yt dt

+dt

from where we obtain under the condition dt 0 the following system of differential equations:

dx = (rx - ry ) xt yt
dt

dy = (r - r ) xt yt y x
dt

xt + yt

(6)

xt + yt

The above system describes the dynamics of densities of X and Y under their age-independent sum-preserving thinning. We see immediately from (6) that when ry > rx , i.e. ry - rx ,>0 (and, correspondingly, rx - ry , < 0) the density of Y permanently increases whereas the density of X permanently decreases. This implies that Y will ultimately supplant X. So, a population with greater intrinsic rates of growth, r, have advantage over that with lesser r and ultimately supplants it. This means that r plays the role of fitness criterion when the competition between the resident and mutants is described by system (6). Fig. 1 illustrates this situation for rx = 0.01 and ry = 0.02. We


Author's personal copy
ecological modelling 2 1 2 (2008) 5­9

7

Fig. 1 ­ The dynamics of densities of populations X and Y in the case of age-independent thinning for rx = 0.01, ry = 0.02, x0 = 20, y0 = 1 and K = 100 (X and Y are marked as resident and invader, correspondingly).

see that asymptotically the number of X approaches to zero whereas the number of Y tends to the initial common density.

2.2.

Thinning the youngs

The parameter R0 can be interpreted as lifetime reproductive success of an individual because it can be viewed as a product of the birth rate b by the inverse of death rate 1/d which, in its turn, is equal to the expected lifespan, L. Though the both criterions, r and R0 , increase with increasing b and decreasing d their maximum values as functions of b and d are attained at different values of these arguments. Hence we may expect that, for some combinations of b and d, an invader Y capable to supplant the resident X in the case of age-independent thinning will be ousted by the resident in the case of thinning the youngs, and vise versa. A typical situation when an invader is more advantageous from the point of view of the criterion r as compared with R0 , arises when great values of r are due to great values of birth rates. For example, if by = 0.10, dy = 0.03, bx = 0.05 and dx = 0.01 then ry - rx = (0.10 - 0.03) - (0.05 - 0.01) = 0.03 but R0y - R0x = (0.10/0.03) - (0.05/0.01) = -1.67. This means that, in the case of an age-independent thinning, Y will inevitably supplant X because ry > rx but if the density press acts only on the youngs then, on the contrary, Y will be supplanted by X because R0y < R0x . That is an invader with a high birth rate but in the same time with a relatively high mortality will win under age-independent thinning and will lose under youngsaimed thinning. We will see in the next paragraph that such an invader will lose in the case of adults-aimed thinning too.

Rewrite now system (1) in another way: xt yt
+dt +dt

2.3.

Thinning the adults

= xt + bx xt dt - dx xt dt = yt + by yt dt - dy yt dt

(7)

and represent the mechanism of proportional reducing the density of youngs as multiplying the second terms of righthand sides of both Eq. (7) by some coefficient (intuitively this process can be imagined as thinning the youngs by a factor of 1/): xt yt
+dt +dt

The thinning of adults can be considered as increasing their death rate in the same proportion both for residents and mutants. To embed this mechanism into the system (7) we will modify it by increasing the death rates in the both equations a factor of i xt yt
+dt +dt

= xt + bx xt dt - idx xt dt = yt + by yt dt - idy yt dt

= xt + bx xt dt - dx xt dt = yt + by yt dt - dy yt dt

After transformations similar to those made in Section 2.1 we obtain:

After transformations similar to those made in Section 2.1 we obtain the following system:

dx = dt dy =
dt

by bx - dx dy by bx - dy dx

dx dy xt yt dx xt + dy yt dx dy xt yt dx xt + dy yt

(9)

dx = dt dy =
dt

by bx - dx dy by bx - dx dy

dx dy xt yt bx xt + by yt dx dy xt yt bx xt + by yt

(8)

This system describes the dynamics of densities of X and Y under sum-preserving thinning of youngs. We see from (8) that when by /dy > bx /dx , i.e. by /dy - bx /dx > 0 (and, correspondingly, bx /dx - by /dy < 0) the density of Y permanently increases whereas the density of X permanently decreases so that Y will ultimately supplant X. So, populations with greater ratios R0 = b/d have advantage over those with smaller R0 that implies that in this model R0 plays the role of fitness criterion. This result contrasts with the situation considered in the previous paragraph where the fitness criterion was not the ratio R0 = b/d of birth rate to death rate but their difference r = b - d.

This system describes the dynamics of densities of X and Y under sum-preserving thinning of adults. Though system (9) slightly differs from (8), we can see that it leads to the same conclusions as (8). Namely, when by /dy > bx /dx , i.e. by /dy - bx /dx > 0 (and, correspondingly, bx /dx - by /dy < 0) the density of Y permanently increases whereas the density of X permanently decreases so that ultimately Y will supplant X. Thus in the situation of thinning adults just as in the previously considered situation of thinning youngs the fitness criterion is the same, R0 .

3.

Logistic limitation

In the models of stopped exponential growth we considered rather rigid ways of preserving the summary population


Author's personal copy
8
ecological modelling 2 1 2 (2008) 5­9

density after having attained the carrying capacity K. The wellknown model of two-population competition (Lotka, 1925; Volterra, 1926; May, 2001) assume a more soft mechanism of stabilizing the summary population density near the carrying capacity K. It looks as follows

dx = rx xt 1 - xt + yt dy = r y 1 - xt + yt yt
dt K dt K (10)

We may call it model of logistic growth because system (10) is a straightforward generalization of well-known model of logistic growth of one population for the case of two populations. What happens if ry > rx ? Will Y supplant X as it was the case in the situation of age-dependent thinning? As we see from Fig. 2 illustrating the dynamics of densities of X and Y, obtained by integrating (10) for rx = 0.01, ry = 0.02, it is not so. In contrast to Fig. 1, Y does not supplant X though ry > rx (note, however, that nor Y supplants X). Thus we may conclude that in the case of logistic model described by system (10) the value of r does not change crucially the fitness of population. But this model has one more phenotypic parameter, the carrying capacity K. If we suppose that this parameter differs for X and Y then we will obtain the following model:

Fig. 3 ­ The dynamics of populations X and Y in the case described by (11) for rx = ry = 0.01, x0 = 20, y0 =1, Kx = 100 Ky = 200 (X and Y are marked as resident and invader, correspondingly).

So we conclude that in the case of logistic environmental press the role of fitness criterion is played by K.

4.

Conclusion

dx xt + yt dt = rx xt 1 - Kx dy = r x 1 - xt + yt
dt
yt

(11)

Ky

An example of dynamics of X and Y, obtained by integrating (11) for Ky = 200 and Kx = 100 for equal intrinsic rates of growth rx = ry = 0.01, is given in Fig. 3. We see that now Y supplants X. The reason is that X, having lesser environment capacity, is more sensible to density stress: as soon as the limit Kx is surpassed the rate of growth of X (determined by the right-hand side of the first equation in (11)) becomes negative whereas the rate of growth of Y remains positive.

Thus we showed that any population formed of invaders or mutants, even very small, that surpasses the resident population in r in the case of age-independent thinning, in R0 in the cases of thinning of only youngs or of only adults, and in K in the case of age-independent steadily increased density press, will ultimately supplant the resident population. Particularly, in the case of recurring mutations this implies that, depending on the type of environmental density stress, successful mutants will have correspondingly higher values of r, R0 or K. So, these characteristics may be considered as criteria of evolutionary fitness for corresponding environmental situations. We may also figuratively say that, depending on the type of environmental density stress, r-, R0 - or K-strategy of evolutionary selection should be realized.

Acknowledgements
The authors are grateful to anonymous referees for helpful suggestions. The work was supported by a grant from the Russian Foundation for Basic Research (07-04-00521).

references

Fig. 2 ­ The dynamics of densities of populations X and Y in the case described by system (10) for rx = 0.01, ry = 0.02, x0 = 20, y0 = 1 and K = 100 (X and Y are marked as resident and invader, correspondingly).

Lotka, A.J., 1925. Elements of Physical biology. Williams and Wilkins, Baltimore. May, R.M., 2001. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton. Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge. Mylius, J., Diekmann, O., 1995. On evolutionary stable life histories, optimization and the need to be specific about density dependence. Oikos 74, 218­224.


Author's personal copy
ecological modelling 2 1 2 (2008) 5­9

9

Teriokhin, A.T., Budilova, E.V., 2001. The evolution of life history: models based on optimization of energy allocation. J. Gen. Biol. 62, 286­295 (in Russian, with English abstract). Teriokhin, A.T., Thomas, F., Budilova, E.V., Guegan, J.F., 2003. The impact of environmental factors on human life-history

evolution: an optimization modeling and data analysis study. Evol. Ecol. Res. 5, 1199­1221. Volterra, V., 1926. Variozioni e fluttuacionie del numero d'individui in specie animali conviventi. Mem. Acad. Lincei. 2, 31­113.