Global Properties of Evolutional Lotka-Volterra System

We will study global properties of evolutional Lotka-Volterra system. We assume that the predatory efficiency is a function of a character of species whose evolution obeys a quantitative genetic model. We will show that the structure of a solution is rather different from that of a non-evolutional system. We will analytically show new ecological features of the dynamics.


Introduction
In this paper, we study global behavior of an evolutional Lotka-Volterra system for three species   for the unknown quantities and 2 which are the population of jth species and the mean character value of the second species, respectively.Here are certain constants, and 2 and 3 are death rate of the second and the third species, respectively.The quantities 2 and 3 are the predatory efficiency of the second and the third species, respectively.The number 2 is the mean character value of the second species with minimal cost.The quantity is the additive genetic variance and   is the cost of evolution, namely, if  decreases, then the cost increases.
The effect of evolution is expressed in terms of (1.4) and the condition that the predatory efficiency   where is a given constant and 0 3 0 a    0 3 0 a z a   is a function of .An example of a 3 is given by (C.1) in Section 3. Equation (1.4) follows the quantitative genetical model (cf.[1][2][3][4][5].See also Section 7).The evolutional Lotka-Volterra system for two species was studied in [3], where rather detailed numerical analysis was made.As for the system for three species, very little is known as to global behavior of solutions even from a numerical point of view.In this paper, we shall make the analytical study of evolutional Lotka-Volterra model for three species and show several new phenomena caused by evolution.We also refer [6] as to non-evolutional case.have unique smooth time global solution.(cf.Theorem 2).Then, in terms of estimate of a solution obtained in the proof of Theorem 2, we study behaviors of a solution related to evolution.Indeed, we will show that the behavior of a solution near the equilibrium point is different from those in the case of tea-cup attractors for a nonevolutional system.Namely, the decay of the predator 3 starts before the quantity 2 becomes small because the predatory efficiency a 3 tends to zero, by evolution.We remark that although N N 3 2 plays an important role in the non-evolutional system near equilibrium point, the quantity N N is crucial in the evolutional one.This is because the quantity 3 2 N N is related with the dynamics of evolution.We remark that the effect of evolution in our system is intermittent in the sense that in some subdomain of the phase space flactuations of pray 1 2 occur as in the case of non-evolutional model, while in other subdomain, evolution stabilizes large fluctuations of 2 and 3 .We also discuss the role of γ in (C.1), which is related with the sensitivity of evolution to the character bias 2 2 .(cf.Lemma 3 and Section 4 for the case of a linear efficiency).In Section 5, we study the uniform convergence of solutions of an evolutional system as the cost of evolution tends to infinity, i.e.,

Time Global Solution
We shall study the global existence and uniqueness of a solution of the initial value problem.We assume that is the twice continuously differentiable function which satisfies for some . Moreover we suppose that there exist and such that The following local existence and uniqueness theorem is well known.
THEOREM 2.1.Assume (2.1) and (2.2).Then there exists a 0   such that the system of Equations (1.1)- (1.4) with the initial conditions (1.6) has a unique In the following we study the existence of a global solution.We require the condition Remark.If for some j, then, by the uniqueness, any solution of (1.1)-(1.4)satisfies Hence it reduces to a system with less unknown quantities.Note that we avoid this case in (2.3).
We have THEOREM 2.2.Suppose that (2. By the local existence and uniqueness theorem, Equations (2.4) with the initial condition has a unique solution.We denote the solution by 0 , , , , 3), we have a contradiction.Hence we have By the continuity of  in a sufficiently small neighborhood of 1 .Then, the second term in the right hand side of (1.1) satisfies where     Note that the right hand side quantity depends on the initial value and the equation and depends neither on δ > 0 nor on g > 0.
We make the same argument for In view of the definition of v we have ,N L  from the below.By the estimates of and from the above there exists such that It follows that   We will estimate N 2 from the below.There exist con-stants 3 3 depending on the equation and the initial values such that, By integrating the inequality from to t we obtain The estimate of from the below can be shown by simple computations.
Next we will prove In order to prove (2.16) we assume that there exists such that either    holds and we show the contradiction.For the sake of simplicity let us assume the former case holds.The latter case can be treated in the same way.By the estimate of   3 N t from the above we have, for any  , (2.17) This is a contradiction.Hence we have the desired estimate.
We shall prove the existence of a global solution.Set where j is the right hand sides of the Equations (1.1)-(1.4),respectively.We write (1.1)-(1.4)into an equivalent system of integral equations Open Access APM By the apriori estimates from the above,  .We will show that it is .For this purpose it is sufficient to show that We note that   2 , j F N z is Lipschitz continuous in each variable because we have apriori estimates of N and 2 .Namely there exists C > 0 independent of N and such that , , Hence, by (1.1)-(1.4)we have This proves the assertion.We can similarly prove for This ends the proof.
Remark. 1) We remark that the apriori estimate of a solution does not depend on the cost of evolution 0     and the additive genetic variance g>0.This means that the evolution of a character has little effect to the bound of sum of populations of three species.
2) As a corollary to Theorem 2 we see that if there is no effect of evolution, i.e.,
The latter equation can be integrated.In view of the uniqueness of the solution of (1.1)-(1.4)we see that (1.1)-(1.3)has a unique solution.

Intermittency of Evolution Effect
We shall study the effect of evolution to dynamics of (1.1)-(1.4).More precisely, we will study how the dynamics of (1.4) is related with that of (1.1)-(1.3).By setting we write (1.4) in the form Let 1   be an integer and and We also assume that   We first study the behavior of 1).Moreover, there exists such that z 0 has an asymptotic behavior ) when .
Proof.We divide the proof into 5 steps.
Step 1.By (C.1) we have and define We consider the zeros and the sign of Step 2. First we consider the case 2.

Assume that
 is an odd integer.Because 1 follows that g  has no zero point on   1,1  .In order to study the zero of g  in (0,1), note .One easily see that the assumption In view of (3.6) we conclude that f   has zero points 0 and 0 in the interval Next we consider the case  is even.By the same way as in the odd case, we have is strictly increasing on (0,1), there exists unique 0 and the argument in the above.
Step 3. We will show the asymptotic formula of 0 in (3.4).In view of the argument in Step 2 we may consider Hence, for sufficiently small we can uniquely solve (3.10).By an implicit function theorem we see that Therefore we have (3.4).

  
Hence we have . Since the right-hand side of (3.12) is bounded for , By solving this relation we have

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By simple computations we obtain (3.5).
Step5.If 1, We can easily see that The rest of the assertion is almost clear from this formula.This completes the proof.
Remark.We will briefly discuss the difference of dynamics of z in (3.1) for f and .
f  We note that two func- tions are identical for 0 .z

  
For a small number 0   , consider the case shown in Figure 2 for 1.

 
Then f looks like as in Figure 3 where new attractive equilibrium points appear near 0


because we have made a modification to 3 so that The new equilibrium point corresponds to that of f  with modulus larger than 0 .Because of the new equilibrium points we have an apriori estimate of the solution for f.Namely, the orbit started from a neighborhood of the origin does not go beyond 0 This fact is important since, if otherwise, the efficiency 3 becomes negative.Note that the dynamics of f and is the same outside some neighborhood of the boundary .We also note that a similar situation occurs in the case 1 and 4).On the other hand, if 0 0     , then the dynamics of f and f  in 0 may be different, while in other part both are the same.

z  z
We also note that the apriori estimate holds for f.Therefore apart from the neighborhood of 0 z   the dynamics of f is well approximated by that of f  , for which f  we can make concrete analysis of the dynam- ics, although we do not have the apriori estimate.We will study behaviors of solutions under the effect of evolution.
1) Behaviors near the equilibrium point.
We consider how the dynamics of (1.1)-(1.4) is related to the dynamics (1.1)-(1.3)without evolution.We recall that (1.1)-(1.3)without an evolutional effect has what is called a tea-cup attractor.The isocline of (1.1)-(1.3),(3.1) is given by the family of equations One may assume that A is small since is small.By (3.14) we have Let us first consider the non-evolutional case, z = 0 or the case where evolution becomes stationary, namely .Then there exists is small and 1 is close to K, by (3.13).It follows that there exists 1 C such that is sufficiently small.In terms of (3.17) and (3.18) .This implies a typical behavior of 2 and 3 around an equilibrium point when there is little effect of evolution.Numerical experiments show that the decrease of 3 occurs soon after the orbit approaches to the equilibrium point, namely N N becomes sufficiently large.
Let us consider the evolutional case.Then the main difference from the non-evolutional case is that   N N grows large.Because 3 2 N z N also grows large, it follows from (3.4) that 0 tends to zero, and we are in the situation that the orbit of 0 tends to 0 .Therefore tends to zero.Hence the boundedness of 2 implies that becomes negative.Therefore, by (1.3), exponentially decreases.Note that the decrease of then evolution progresses.(cf. Figure 2 and Figure 3).The latter condition means either the cost of evolution is small, 1 1 N N  .We note that the attracttive equilibrium points near 0 have the effect to hold the orbits around 0 .Conversely, if , then we see that fluctuations in progress and rest of evolution takes place.
In the case 2 then the evolution becomes stationary.If otherwise, then similar fluctuations in progress and rest of evolution as in the case 1   takes place.We will show in the next section that in the linear case . 3) Fluctuations of and .
1 2 The rhythm of 1 and 2 is also observed in a non-evolutional system and it is related with the structure of a tea-cup attractor.We have a similar phenomenon for an evolutional system.

N N
Let 1 2 3 and be a solution of (1.1)-(1.4).One can show by Poincaré -Bendixon theorem that 1 and 2 are an oscillating solution of two species under appropriate choice of parameters.Note that tends to zero exponentially.By the continuity of solutions of the initial value problem with respect to an initial value and the apriori estimate of a solution, one can see that for every and , , 0


for all Here, without loss of generality we may assume that the initial time is 0.Especially, this shows that there appears a rhythm of 1 and 2 for some interval of time.Note that is small and the evolution becomes stationary, i.e., .
In order to estimate T, we take .By the same reasoning as in 1) 3 tends to zero.We note that in the limit case z  0 a  N    the third species dies out.This agrees with an ecological observation.

Evolution for a Linear Predatory Efficiency
We will discuss the evolution in the case where 0   is a real constant and 0 .As in the previous case we make modifications of in some small neighborhood of the zero point For the sake of simplicity, we assume that 0 . By repeating the same arguments as in Section 2 we see that the system of Equations (1.1)-(1.4)with the initial condition (1.6) has a unique global solution in 0 We will study the dynamics of the evolution in relation with the populations and .We now define 2

.2)
The condition   0 By definition we may consider (4.3) in the set because, if otherwise, We recall that then there appear an attractive equilibrium point 0 near the origin .This means that the predatory efficiency 3 is close to a constant function if N N is sufficiently small.Indeed, the equilibrium point can be estimated as N N .Suppose now that (4.4) does not hold.Then the attracttive equilibrium point near 0 z  disappears, and there remains an attractive equilibrium point near the zero of 3 .Hence the evolution progresses and tends to zero.This alternative between the rest and the progress of evolution shows a high contrast to the case of a convex predatory efficiency function discussed in the previous section.

Behaviors of Solutions as Cost Increases
In this section we study the convergence of a solution of an evolutional system to that of a non-evolutional one when the evolutional cost increases, namely  decreases to zero.Let    , Then we have If we make the change of variables,     z z z   , g and w be the average character value,the additive genetic variance and the average adaptability of the character value z, respectively.Following the quantitative genetical model we have (cf.[1] and [2]).
The left-hand side is the speed of evolution of a character value.Following Fisher, [7] we have e .
r W  We assume that (cf.[3][4][5]) where  is the cost of evolution.By definition we have Because one may regards j as a constant function when z varies, the right-hand side of (7.1) can be replaced by 3) with the initial condition (1.6) has a unique global solution.Indeed, (1.1)-(1.4)can be split into (1.1)-(1.4),

3 a
z is twice continuously differentiable and nonnegative in the closed intervals   .If we denote the right-hand side of (3.1) by by (C.1) we have

Figure 3 .Figure 4 .
Figure 3. Picture of f. . 1 γ  may tends to zero.For the sake of simplicity, let us con-


sents sensitivity to z. Namely, as  increases, z approaches to a constant function.The dynamics of z is quite different in the cases   and 2

THEOREM 5 . 1 .
Assume (2.3).Let be arbitrarily given.Then we have â z z value of a vector means the norm of a vector.Because we have the uniform estimate of   , N s  in  by (1) of Remark in Section 2, we have we see that g  has a unique zero point 0 For the sake of simplicity, we consider the case .Denoting the right hand side of the inequality by v(t) we have relations, We will briefly show how to deduce (1.4) from the theory of quantitative genetics.Let