Single species evolutionary dynamics

Transcription

1 Evolutionary Ecology 17: 33 49, Ó 2003 Kluwer Academic Publishers. Printed in the Netherlands. Research article Single species evolutionary dynamics JOSEPH APALOO Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, P.O. Box5000, Antigonish, Nova Scotia Canada, B2G 2W5 ( Received 12 March 2001; accepted 3 October 2002 Co-ordinating editor:j.a.j. Metz Abstract. Not long after the introduction of evolutionary stable strategy (ESS) concept, it was noticed that dynamic selection did not always lead to the establishment of the ESS. The concept of continuously stable strategy (CSS) was thereafter developed. It was generally accepted that dynamic selection leads to the establishment of an ESS if it is a CSS. Examination of an evolutionary stability concept which is called neighborhood invader strategy (NIS) shows that it may be impossible for an ESS to be established through dynamic selection even if it is a CSS and no polymorphisms occur. We will examine the NIS concept and its implications for two evolutionary game models:root-shoot allocation in plant competition and Lotka Volterra competition. In the root-shoot model we show that an ESS will be attained through dynamic selection if it is a NIS. Similarly for the Lotka Volterra model, we show that an ESS will be attained through dynamic selection even if protected dimorphisms occur during the evolutionary process if it is an NIS. Key words: convergence stability, continuously stable strategy, evolutionary stable strategy, neighborhood invader strategy Introduction The concept of continuously stable strategy (CSS) was first introduced by Eshel and Motro (1981) (also see Eshel, 1983; Taylor, 1989; Lessard, 1990; Nowak, 1990; Christiansen, 1991; Kisdi and Mesze na, 1995; etc.). This concept was defined for evolution when pure strategies admit a continuum of values. CSS is defined in terms of the concept of evolutionary stable strategies (ESS) (Maynard Smith and Price, 1973). An ESS is a strategy such that, if most of the members of the population adopt it, there is no mutant strategy that would have a higher reproductive fitness. (Maynard Smith and Price, 1973). A strategy u is said to be m-stable (convergence stable) if, whenever the entire population has a strategy which is close enough to it, there will be a selective advantage to some individual strategies which are closer to u (Eshel and Motro, 1981; Taylor, 1989; Christiansen, 1991). A CSS is an ESS that is m-stable (Eshel and Motro,

2 ). See Lessard (1990) for a review of several evolutionary stability concepts. It has been noted that an ESS is not an automatic outcome of dynamic process of natural selection (Eshel and Motro, 1981; Maynard Smith, 1982, etc.). Eshel and Motro (1981) showed that CSS are the only class that represents a possible dynamic selection process which eventually leads to the establishment of a CSS in the population. In the discussion section of their paper, they conclude that a small perturbation of an ESS will be followed by a dynamic selection process which will lead to a restoration of the ESS if it is a CSS (see also e.g. Kisdi and Mesze na, 1995). In a recent article (Apaloo, 1997a) the concept of evolutionary stable neighborhood invader strategy (ESNIS) was analyzed for single species evolution. A component of this concept is the neighborhood invader strategy (NIS) concept. We define a NIS as a strategy that is able to invade any established community at ecological equilibrium using any strategy that is in a sufficiently small neighborhood of the NIS (McKelvey and Apaloo, 1995; Apaloo, 1997a, b) (also see Lessard, 1990; Ludwig and Levin, 1991; Kisdi and Mesze na, 1993, 1995). If an ESS is also a NIS, then we refer to it as an ESNIS. It was proven in Apaloo (1997a) that the CSS and ESNIS concepts are not equivalent. This result is easily seen to hold since an ESNIS displaces any of its near neighbors in pairwise ecological competition. The reason for this is that an ESNIS is not invadable by its near neighbors and it can invade any of its near neighbors. A CSS on the other hand may not have this capability since it may not be able to displace its near neighbors in pairwise ecological competitions. The mathematical conditions for CSS only require that some phenotypes in a closer neighborhood of the CSS be able to invade a phenotype that is farther away from the CSS. We note that invasion does not necessarily mean displacement. In fact this invasibility condition is not even required of the CSS itself. It was also shown in particular that a phenotype that is a CSS is not necessarily an ESNIS, but an ESNIS is a CSS. Numerical examples are also given in Apaloo (1997a). Analysis of a simple microevolutionary process indicates that the dynamic process of natural selection leads to the establishment of the ESNIS in a single evolving species if no polymorphisms occur during the evolutionary process, a conclusion which may not hold for a CSS. Predicting the outcome of evolution through dynamic selection of traits is a very important and interesting topic. For the most part, the outcome is taken to conform to the ESS of the corresponding evolutionary game. However, it is well known now that an ESS may not be the outcome of evolution through dynamic selection process (Apaloo, 1997a; Geritz et al., 1998; references therein). But predicting the true outcomes is not a simple process.

3 A recent detailed discussion of how complicated the dynamics of evolution is in the vicinity of an evolutionary singular point has been given by Geritz et al. (1997, 1998) and Kisdi (1999). These authors introduce the concept of evolutionary singular strategy as a generalization of the ESS concept and they define this point to be a point at which the local fitness gradient is zero. Geritz et al. (1998) identified four evolutionary stability classifications of an evolutionary singular point. These classifications are ESS-stability, convergence stability, the ability of a singular strategy to invade other strategies in its neighborhood if initially rare itself, and the possibility of protected dimorphisms. The ability of a singular strategy to invade other strategies if initially rare itself has been called NIS by Apaloo (1997a, b). Using these classifications and the idea of pairwise invasibility plots (Van Tienderen and De Jong, 1986), Geritz et al. (1998) (also see Dieckmann, 1997) devised a nice and simple symmetric graphical scheme which completely classifies all different possibilities for the invasibility of a community by mutants close to an evolutionarily singular point (see Figure 2 in Geritz et al., 1998). These classifications provide a general framework for modeling adaptive trait dynamics (Geritz et al., 1997, 1998; Kisdi, 1999). Geritz et al. (1998) also showed that in the case where a singular strategy is an ESS and convergence stable, i.e. CSS, the population gradually evolves towards the ESS in the long run. But we know that a CSS may not be an ESNIS and therefore it may not be able to invade and displace incumbent communities using strategies in the neighborhood of the CSS. Indeed this conclusion should also hold for some strategies that are arbitrarily close to the CSS. If an ESS is restored after a small perturbation of the ESS through the force of natural selection, then we say that the ESS possesses dynamic stability. This definition is also applicable to the situation where evolution begins with a strategy in the close neighborhood of an ESS. We wish to contribute to the discussion on trait dynamics by examining evolutionary dynamics in the case when the ESS is also an NIS (an ESNIS). In Figure 2 (Geritz et al., 1998) the classifications in the first and fourth quadrants are NISs and those in the fourth quadrant are ESNISs. More specifically, we will examine the question of dynamic stability of an ESS. We use the requirements of an ESNIS together with an Index Theorem (Hofbauer and Sigmund, 1988) in the trait dynamics discussion. In Section 2 we review necessary evolutionary stability concepts. Section 3 will discuss dynamic stability of an ESS when dimorphisms occur and when no dimorphisms occur. In Section 4, two evolutionary game models will be analyzed with the theoretical results obtained in Section 3. The two models are the Lotka Volterra competition and root-shoot allocation in plant competition models. Some concluding remarks are given in Section 5. 35

4 36 Necessary and sufficient conditions for an ESNIS In this section we review the necessary and sufficient conditions for an ESNIS. These will be used in the discussion on trait dynamics. Consider a single species whose ecological dynamics is given by 1 dx ¼ Gðu; xþ x dt ð1þ where x is the species biomass and u is its strategy chosen from a one-dimensional continuum. We assume that Equation (1) has stable steady state solutions for each u in a close neighborhood of strategy u that is to be examined for evolutionary stability. Let gðv; u; x^þðgðv; uþ for short) denote the per-capita growth rate of rare mutants with strategy v entering a community of individuals at ecological equilibrium x^ using strategy u. This is essentially the fitness generating function introduced by Brown and Vincent (1987). Note the convention here that the first argument of g is the entrant s phenotype and the second argument is the incumbents phenotype and note also that x^ ¼ x^ðuþ. Initial ecological dynamics of a rare entrant with phenotype v is 1 dy ¼ gðv; u; x^ Þ y dt Thus v can invade or cannot invade u according as gðv; u; x^þ > 0 or gðv; u; x^þ < 0. With the above notation we provide the following definitions which are to be interpreted in the local sense. Definition 1. A strategy u is said to be an ESS if gðu; u ; x^þ < 0, for any u in a small neighborhood of u, u 6¼ u, and an NIS if gðu ; u; x^þ > 0 for any u in a small neighborhood of u, u 6¼ u (Apaloo, 1997a). The NIS and ESS definitions can be interpreted as extreme values of the respective functions gðu ; u; x^ Þ and gðu; u ; x^þ. Thus the following are the sufficient conditions for ESS and NIS (Apaloo, 1997a). We note that an equivalent statement of this result has been given in Metz et al. (1996), Geritz et al. (1997, 1998) and Kisdi (1999). Theorem 2. If u is an interior point of the space of strategies and gðv; uþ is a smooth function in v and u, then sufficient conditions for u to have the above evolutionary stabilities are: ESS : NIS : ogðv; u Þ ov ogðu ; uþ ou ¼ 0 and o2 gðv; u Þ ov 2 < 0at v ¼ u ¼ 0and o2 gðu ; uþ ou 2 > 0at u ¼ u

5 Let a fixed phenotype u be within neighborhood of u. In the diagram below, a line segment marked with = indicates a set of strategies that can invade u and a line segment indicated with x denotes a set of strategies that cannot be invaded by u. With the above notation, we obtain the following geometric illustrations of the nature of interaction between u and phenotypes that are closer to u, when it is an ESNIS, assuming that the fitness functions are smooth. We note that these diagrams are obtained from the definitions of the ESNIS concept in the following manner. If u is an ESNIS then it cannot be invaded by its near neighbors since it is an ESS. Thus u cannot invade u or phenotypes in an arbitrary small neighborhood of u denoted by x in the diagram below (by continuity). If u is an ESNIS then it can invade u since it is a NIS, and by continuity, phenotypes in an arbitrary small neighborhood of u can invade u. This set of phenotypes that can also invade u when u is an ESNIS is denoted by = in the ESNIS diagram. 37 So u is an ESS if it lies in an x set, an ESNIS if it is an ESS and it lies in an ¼ set. We note here that when u is a CSS, u may not lie in the = set. For further discussions on the differences between an ESNIS and a CSS (see Metz et al. 1996; Apaloo, 1997a; Geritz et al. 1997, 1998; Kisdi 1999). Dynamic stability In this section we will examine the question of dynamic stability of an ESS of a single species evolutionary model. It is well known that two strategies on opposite sides of an ESNIS in a monomorphic population evolution may maintain stable coexistence (protected dimorphism) (see for example Metz et al., 1996; Apaloo, 1997a; Geritz et al., 1997, 1998; Kisdi, 1999). In the absence of any protected polymorphisms, dynamic selection will lead to the establishment of the ESNIS (Apaloo, 1997a). The case where protected dimorphisms may occur during the evolutionary process is of particular interest in the remainder of this article. We recall that in such a case, the arrival of a third subspecies will result in a three-dimensional ecological interaction. Geritz et al. (1998) note that if an ESS is also convergence stable, then no more than two strategies in the neighborhood of the CSS can maintain stable coexistence. Therefore the arrival of a new mutant encountering a dimorphic population results in a dynamic of three strategies with only one of the following possible outcomes; mutant may be repelled, or mutant displaces both incumbents or mutant coexists with only one of the two incumbents (Geritz

6 38 et al., 1998). Christiansen and Loeschke (1987) showed that in the neighborhood of an ESS in one-dimensional trait space, trimorphisms are generally excluded (also see Eshel et al., 1997). We therefore assume in the results below that at any ecological equilibrium involving strategies, there can only be at most two strategies. In view of the fact in the preceding paragraph and the knowledge that an ESNIS will always displace any monomorphic strategy, we now focus on the situation in which protected dimorphisms occur and investigate the outcome of the dynamic process if an ESNIS is introduced into a stable dimorphism using strategies that are in the neighborhood of the ESNIS. Let us for now assume that a three-dimensional community with three distinct strategies denoted by the vector u ¼ðu 1 ; u 2 ; u 3 Þ, one of which, u 1, is the ESNIS u, cannot coexist. In particular, if the vector x denotes the population sizes of the phenotypes, we are assuming that the dynamical system 1 dx i x i dt ¼ f iðu i ; u; xþ i ¼ 1; 2; 3 ð2þ modeling the dynamic interactions between the three species has no interior equilibrium point. It is the case then that there are no limit cycles which may be seen from the following result due to Hofbauer and Sigmund (1988). Let R n þ ¼fx¼ðx 1; x 2 ;...; x n Þ 2 R n : x i 0 for i ¼ 1; 2;...; ng. Theorem 3. The interior of R n þ contains a or x limit points if and only if the above system admits an interior equilibrium (Theorem 1 on page 60). Definition 4. Let U be a bounded open subset of R n and f a vector field defined on a neighborhood of its closure U. A point x 2 U is said to be regular if det D x f 6¼ 0 where D x f is the Jacobian matrix of f evaluated at x 2 R n.a point y 2 R n is said to be a regular value if all x 2 U with fðxþ ¼y are regular (Hofbauer and Sigmund, 1988, page 162). Definition 5. If x^ is regular, then the Poincare index of x^, iðx^þ is given by iðx^ Þ¼ð 1Þ r where r is the number of real negative eigenvalues of the Jacobian matrix D x^f (Hofbauer and Sigmund, 1988, page 163). Definition 6. A rest point x^ of (2) is said to be saturated if f i ðx^þ 0 when x^ i ¼ 0 and f i ðx^ Þ¼0 when x^ i > 0 (Hofbauer and Sigmund, 1988, page 166). Theorem 7. (Index Theorem for system (2)). If the system (2) has uniformly bounded orbits then it has a saturated fixed point, and if all saturated fixed points are regular then the sum of their indices is ð 1Þ n (Hofbauer and Sigmund, 1988, page 167, exercise 3; also see page 167, exercise 4).

7 In what follows, we will denote the equilibrium population density at the monomorphic ESNIS u by x^ ¼ x^ðu Þ. As noted above, we will take the strategy u 1 in the system of equations (2) as the ESNIS u and let x^ ¼ ðx^ ; 0; 0Þ be the monomorphic ESNIS equilibrium in the three-dimensional system (2). Claim. Suppose that all equilibrium points of the system (2) are regular and that the system has no interior equilibrium point. Then any boundary equilibrium point of (2) other than the equilibrium point x^ has a positive eigenvalue with corresponding eigenvector in the direction parallel to the x 1 - axis, i.e. all boundary equilibrium points of (2) other than the equilibrium point x^ are not saturated. Proof. Assume that all equilibrium points of the system (2) are regular and that the system has no interior equilibrium point. We want to show that any boundary equilibrium point of (2) other than the equilibrium point x^ has a positive eigenvalue with corresponding eigenvector in the direction of x 1, i.e. all boundary equilibrium points of (2) other than the equilibrium point x^ are not saturated. First we note that x^ is saturated since it is the equilibrium point corresponding to an ESS. We also note that the equilibrium point ð0; 0; 0Þ, and the monomorphic equilibrium points ð0; x^ 2; 0Þ and ð0; 0; x^ 3Þ which correspond to the strategies u 2 and u 3 respectively are not saturated since u is an ESNIS. In addition, the dimorphic equilibrium points ð~x 1 ; ~x 2 ; 0Þ (for strategies u and u 2 ) and ð~x 1 ; 0; ~x 3 Þ (for strategies u and u 3 ) do not exist since u is an ESNIS. Thus the only questionable equilibrium point is the dimorphic equilibrium point ð0; ~x 2 ; ~x 3 Þ corresponding to the strategies u 2 and u 3. Suppose that the latter dimorphic equilibrium point is saturated and thus has index -1. Also the stable equilibrium x^ has index -1. So the sum of the indices of the saturated and regular equilibria is 2. But by the index theorem, the sum must be 1 which is a contradiction. Thus the dimorphic equilibrium point ð0; ~x 2 ; ~x 3 Þ is not saturated. The consequence of the above claim is that the equilibrium point x^ is globally stable if there are no interior equilibrium points and all equilibrium points are regular. In this section we have noted that for single species evolution, the outcome of evolution through dynamic selection is an ESNIS if the ecological stable phases of the evolutionary process involves no more than one strategy. In the case where dimorphisms occur in the stable ecological phases of dynamic evolutionary process, we have shown that the outcome of evolution will be an ESNIS in the following sense. In any three-dimensional system involving the ESNIS, the ESNIS equilibrium population size when present alone is globally stable when no interior equilibrium occurs and all equilibrium points are regular. We give two examples of these results in the next section. 39

8 40 Examples In this section we present two examples to illustrate the two results discussed in the previous section. Lotka Volterra competition We will now discuss some of the implications of the ESNIS concept for the evolution of strategies when their ecological interactions are modeled by the Lotka Volterra competition (see e.g. Brown and Vincent, 1987; Metz et al., 1996; Kisdi, 1999). We begin by first considering the single species ecological equation 1 dx x dt! ^ x ¼ Gðu; xþ ¼r 1 kðuþ Suppose that the population is at stable equilibrium x^ðuþ. An entrant to this community with strategy v will have fitness aðv; uþx gðv; u; x^ Þ¼r 1 kðuþ We choose the carrying capacity and competitive interaction functions KðuÞ ¼100 exp 1! u 2 2 r k and aðv; uþ ¼1 þ exp 1! v u þ b 2 exp 1! b 2 2 r a 2 r a with b ¼ r a ¼ r k ¼ 2 (Brown and Vincent, 1987). Applying the conditions for ESNIS above, it can easily be shown that an ESNIS u ¼ 1:213 exists for this single species model for the above choice of functions and parameter values. Since an ESNIS is a CSS, it follows that u ¼ 1:213 is a CSS. In addition, from Table 1 in Apaloo (1997a), we see that protected dimorphisms can occur in the neighborhood of u ¼ 1:213. Therefore this strategy has the configuration (c) in Figure 2 in Geritz et al. (1998). As done above we adopt the notation that the population size of the ESNIS u (¼ u 1 ) will always be denoted by x 1 ðtþ. Let x ¼ðx 1 ; x 2 ; x 3 Þ as we keep in mind that arrival of an ESNIS to a protected dimorphism is the case of interest here. The ecological dynamics between u and two other strategies u 2, and u 3 in the neighborhood of u is given by the three-dimensional system of equations

9 41 dx i dt ¼ rx i kðu i Þ kðu i Þ Xn j¼1 aðu i ; u j Þx j! ¼ x i f i ðu; xþ; i ¼ 1; 2; 3 Recall that the equilibrium population density at the monomorphic ESNIS u is denoted by x^ ¼ x^ðu Þ and x^ ¼ðx^ ; 0; 0Þ is the monomorphic ESNIS equilibrium in the three-dimensional system. The question here is whether the equilibrium point x^ is the globally stable equilibrium point of this threedimensional system. We explore this question below. First we note that unique positive equilibrium point (x 1; x 2; x 3Þ (x i > 0 for each i) does not exist for the above system (2). This conclusion can be arrived at as follows. The system of equations that can be solved to obtain this interior equilibrium point is given in matrix form as a 12 a 13 a 21 1 a 23 a 31 a x x 2 x ¼ 4 k 1 k 2 k 3 where a ij ¼ aðu i ; u j Þ, k i ¼ kðu i Þ and we refer to the 3 3 matrix as the a matrix. For this system to have unique positive solution (i.e. x i > 0 for each i) the following determinants must all have the same sign: k 1 a 12 a 13 1 k 1 a 13 k 2 1 a 23 k 3 a 32 1 ; 1 a 12 k 1 a 21 k 2 a 23 a 31 k 3 1 ; 1 a 12 a 13 a 21 1 k 2 a 31 a 32 k 3 ; a 21 1 a 23 a 31 a 32 1 Here we are excluding the degenerate case where the last determinant is zero. A biologically meaningful form of the first three determinants can be obtained by expanding them by the minors around the ith row (Strobeck, 1973). In the case where the determinants are positive the conditions reduce to k i > ð<þ P a ij x ðiþ j if D ðiþ > ð<þ0 for all i, where x ðiþ j is the equilibrium number of the jth species when the ith species is not present and D ðiþ is the determinant of the a matrix without the ith species. For the above model, in particular for the specific choice of the a ij ¼ aðu i ; u j Þ above, D ðiþ > 0 for all i. Thus the conditions for unique positive equilibrium further reduce to k i > P a ij x ðiþ j for each i. This condition is valid for i ¼ 1 since u when rare can invade a protected dimorphism in the neighborhood of an ESNIS (Geritz et al., 1998). But the inequality is reversed when i ¼ 2 since u 2, when rare cannot invade the community ðu ; u 3 Þ since this community will entirely consist of members using the strategy u as u is an ESNIS. It is very important to note that the conclusion may not hold if u is just an ESS or a CSS. Therefore the conditions for unique positive equilibrium is violated and thus a unique positive equilibrium does not exist for such systems when u is an ESNIS. 5

10 42 It can easily be verified that all possible equilibrium points (as listed in the Claim above) are regular. From the discussion above we know that the above system has no interior equilibrium point and so the system (2) admits no interior x-limit points (i.e. has no limit cycles or periodic orbits) as discussed earlier. Thus x-limit points must lie on the boundary and the simple dynamics on the boundary implies that such x-limit points are equilibrium points. It therefore follows that the basin of attraction (or region of asymptotic stability) of x^ is the interior of R n þ since there is no interior equilibrium and the only questionable equilibrium ð0; ~x 2 ; ~x 3 Þ has positive eigenvalue with corresponding eigenvector in the direction of x 1 (see Dynamic stability section above). Thus x^ is globally asymptotically stable point of the system (2). Thus we have shown that for the Lotka Volterra evolutionary model given above, a dimorphism composed of members using strategies on the opposite sides of the ESNIS u will be displaced by the ESNIS. This conclusion is also evident in the phase portrait 7 on page 202 in Zeeman (1993). We therefore conclude that if an ESS is also an NIS (i.e. an ESNIS) then not only does the population gradually evolve towards the ESS but the ESS is attainable through dynamic selection of strategies even if protected dimorphisms occur. Root-to-shoot ratio and plant competition In this section we consider the case where the ecological phases of the evolutionary process involves no more than one strategy. As noted above, evolution will lead to the establishment of the ESNIS since the arrival of an ESNIS will displace any existing incumbent. In addition strategies closer to the ESNIS will be favored in competition with those farther away from the ESNIS. This latter conclusion is easily seen as an ESNIS is a CSS. Vincent and Vincent (1996) applied strategic evolutionary game techniques to examine competition and coexistence in root-shoot allocation in plant growth under different resource conditions. The model they used is a modified version of the model developed by Reynolds and Pacala (1993). We refer the reader to these papers for the details. The model used is 1 db i B i dt ¼ min rnu i R; rlð1 u iþ N þ k N L þ k L dn dt ¼ a Xn T N L 0 i¼1 pb i! L ¼ 1 þ P n i¼1 ab ið1 u i Þ Xn i¼1 R d; i ¼ 1; 2;...; n rnu i min R; rlð1 u iþ R pb i N þ k N L þ k L

11 where B i is the biomass of population i, N the available soil nutrient, L the light availability, u i the fraction of biomass allocated to root by population i, r the per-capita maximal rate of plant growth, k N the half-saturation constant for nutrient-limited plant growth, k L the half-saturation constant for light-limited plant growth, R the density independent per-capita respiration rate, d the density independent per-capita loss rate, T the total amount of all forms of soil nutrient in a particular habitat, p the plant tissue nutrient concentration, a the light decay rate per unit of leaf biomass, L 0 the solar constant, a the mineralization rate. Consider the above system with n ¼ 1, then the fitness of an invader with strategy v in an environment determined by u be given by gðv; u; N; LÞ ¼min rnv rlð1 vþ R; R d N þ k N L þ k L as discussed above. Note that the equilibrium values of both N and L will depend on u. It is important to note here that the fitness function here is not smooth and so we use only the definitions of the evolutionary stabilities (Definition 1) and their graphical interpretations in the discussion below, and we do not use Theorem 2. In a bare habitat, it can be shown that only plants with fraction, u, of biomass allocated to root lying in the interval (Reynolds and Pacala, 1993) d þ R r 1 þ k N T ; 1 d þ R r 1 þ k L L 0 can sustain initial increase in the environment. Thus one only need search this interval for candidate ESS and or NIS. We also note that when an incumbent species is not adopting the ESS, a second species can invade only if the second species is slightly less limited by the same resource that the incumbent is limited by (Reynolds and Pacala, 1993). Indeed in this case, the species that is slightly less limited will win the competition. Thus coexistence is impossible in this model and the ecological phases of the evolutionary process involves no more than one strategy. It is clear that if an ESNIS exists for this model, then dynamic selection will lead to the establishment of the ESS. Following Vincent and Vincent (1996) we make the following specific assumptions on the parameter values: a ¼ 0:3, r ¼ 5, k N ¼ k L ¼ 1, p ¼ 0:1, R ¼ 0:5, a ¼ 0:001, L 0 ¼ 2, T ¼ 5, d ¼ 0:5. Vincent and Vincent (1996) obtained the point u ¼ 0: with the equilibrium environment B ¼ 17:2515, N ¼ 0:399587, and L ¼ 2: For the specific values of the parameters chosen above, the interval for positive initial growth in a bare environment is ð0:24; 0:70Þ. Thus the strategy obtained in Vincent and Vincent (1996) cannot support a positive equilibrium and is therefore not an ESS for this model. Solving the same equations we obtain the corresponding values u ¼ 0:6991 with the equilibrium environment B ¼ 17:2307, N ¼ 0:4010, and L ¼ 1:99 43

12 44 which is in better agreement with the simulation results in Vincent and Vincent (1996) than above results. We analyzed the model to ascertain whether the candidate ESS u ¼ 0:6991 (with the equilibrium environment B ¼ 17:2307, N ¼ 0:4010, and L ¼ 1:99) is an ESNIS or not. The analysis reveals that there cannot be an arrested succession if evolution is confined to the left-neighborhood of u and this agrees with Vincent and Vincent (1996) and is in disagreement with Reynolds and Pacala (1993). Reynolds and Pacala (1993) denoted the final strategy at the end point of species successional sequence by A f where succession here refers to the process where new species become dominant. They showed that species with allocation strategy 0:45 could not be invaded by those with the allocation strategy A f ¼ 0:67 due to the fact that these two species are founder controlled. Since we do not know the value of the parameter d which was used in Reynolds and Pacala (1993), we will take the final dominant strategy to be the strategy u ¼ 0:6991 given above. Results of simulation of the two strategies u ¼ 0:6991 and 0:45 is given in Figure 1 from which we see u can invade and displace 0:45. Thus the final dominant strategy and the strategy 0:45 are not founder controlled. Indeed the strategy u ¼ 0:6991 is an ESNIS with respect to strategies in its left-neighborhood and so evolution beginning with a strategy in the left-neighborhood of u ¼ 0:6991 with subsequent mutations confined to the same interval will lead to the establishment of u ¼ 0:6991 through dynamic selection. Figure 1. Outcome of competition between two species using the strategies u ¼ 0:6991 and u ¼ 0:45 over time. The biomass of strategy u is B 2 and the biomass for strategy u is B 1 with respective initial population sizes 2 and

13 45 Figure 2. Outcome of competition between two species using the strategies u ¼ 0:6991 and u ¼ 0:6994 over time. The biomass of strategy u is B 2 and the biomass for strategy u is B 1 with respective initial population sizes and 2. Figure 3. Outcome of competition between two species using the strategies u ¼ 0:6991 and u ¼ 0:6995 over time. The biomass of strategy u is B 1 and the biomass for strategy u is B 2 with respective initial population sizes 2 and The other important revelation is that the above candidate ESS is not an ESS nor an NIS with respect to strategies in its right-neighborhood. This means that if evolution begins with a strategy in the right-neighborhood of u ¼ 0:6991, dynamic selection may not lead to the establishment of the strategy u ¼ 0:6991.

14 46 Figure 4. Outcome of competition between two species using the strategies u ¼ 0:6994 and v ¼ 0:6995 over time. The biomass of strategy u is B 1 and the biomass for strategy v is B 2 with respective initial population sizes 2 and Figure 5. Graphs of fitnesses gðu; u Þ and gðu ; uþ with u ¼ 0:6991. The nature of these graphs indicate that is an not ESNIS. Thus arrested evolution can occur if evolution were to begin with strategy in the right-neighborhood of u ¼ 0:6991. This result can be inferred from Figures 2 4 which show that an allocation strategy u þ <7 (for some >0) can

15 47 Figure 6. Graphs of fitnesses gðu; u Þ and gðu ; uþ with u ¼ 0:6994. The nature of these graphs indicate that is an ESNIS. displace the strategy u. This result is a consequence of the fact that the above u is not an NIS. The fact that u is not an NIS nor an ESS can be seen in Figure 5 as gðu ; uþ is not >0 for all u 6¼ u and gðu; u Þ is not <0 for all u 6¼ u respectively. Figures 2 6 suggest that the strategy 0:6994 (to four decimal places) is an ESNIS. Such a strategy will be the outcome of evolution through dynamic selection and in this case arrested evolution cannot occur. Discussion In this article we have examined dynamic stability of an ESS for single species evolution in the case where at most protected dimorphisms occur in the evolutionary process. The results were applied to species competition when the strategies interact according to the Lotka Volterra competition, and to the root-to-shoot ratio and plant competition for soil nutrient and light. With the help of the concept of ESNIS, we have refined the prediction of the evolutionary outcome of the root-to-shoot allocation problem. We also showed that arrested evolution cannot occur when the root-to-shoot allocation ratio problem possesses an ESNIS. This model is an example of single species evolution in which the stable ecological phases of evolution involve only one strategy for which it is well known that the outcome of dynamic selection will be an ESNIS and not necessarily an ESS or CSS. In the second example in which we consider evolution in which ecological interaction is modeled by Lotka Volterra equations, we have shown that an

16 48 ESNIS for the Lotka Volterra evolutionary model cannot coexist with any finite number (up to two) of strategies in its neighborhood. In particular we showed that such an equilibrium state does not even exist. Secondly, we have shown that the equilibrium population when the ESNIS is present alone is globally stable in the n-dimensional system (for n ¼ 1; 2; 3) in which u interacts ecologically with n 1 other strategies in its neighborhood. The consequence of this is that an ESNIS for the single species evolution will be established through dynamic selection even if protected dimorphisms occur. These conclusions cannot be assumed to hold if the strategy is a CSS. It is clear then that the ESNIS concept can play a very useful role in understanding evolutionary dynamics of strategies. Further research needs to be done to explore the implications of this concept for other evolving strategies whose ecological interactions are not of the Lotka Volterra type and for coevolution. Acknowledgements The author wishes to thank anonymous reviewers and Dr G. Mesze na for helpful comments, and Dr J.A.J. Metz especially for critical comments that helped bring more clarity to this manuscript. Financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada, Saint Francis Xavier University Starter Grant and University Council for Research (UCR), and the University of Natal University Research Fund (URF) is gratefully acknowledged. References Apaloo, J. (1997a) Revisiting strategic models of evolution:the concept of neighborhood invader strategies. Theor. Popul. Biol. 52, Apaloo, J. (1997b) Ecological Species Coevolution. J. Biol. Syst. 5, Brown, J.S. and Vincent, T.L. (1987) Coevolution as an evolutionary game. Evolution 41, Christiansen, F.B. (1991) On conditions for evolutionary stability for continuously varying character. Am. Nat. 138, Christiansen, F.B. and Loeschcke, V. (1987) Evolution and intraspecific competition. III. Onelocus theory for small additive gene effects and multidimensional resource qualities. Theor. Popul. Biol. 31, Dieckmann (1997) Can adaptive dynamics invade? TREE 12, Eshel, I. (1983) Evolutionary and continuous stability. J. Theor. Biol. 103, Eshel, I. and Motro, U. (1981) Kin selection and strong evolutionary stability of mutual help. Theor. Popul. Biol. 19, Eshel, I., Motro, U. and Sansone, E. (1997) Continuous stability and evolutionary convergence. J. Theor. Biol. 185, Geritz, S.A.H., Kisdi, E., Mesze na, G. and Metz, J.A.J. (1998) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12,

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