ABSTRACT

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.

1. Introduction

In this paper, we study global behavior of an evolutional Lotka-Volterra system for three species

(1.1)

(1.2)

(1.3)

(1.4)

for the unknown quantities and which are the population of jth species and the mean character value of the second species, respectively. Here are certain constants, and and are death rate of the second and the third species, respectively. The quantities and are the predatory efficiency of the second and the third species, respectively. The number 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 is given by

(1.5)

where is a given constant and is a function of. An example of a₃ is given by (C.1) in Section 3. Equation (1.4) follows the quantitative genetical model (cf. [1-5]. 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.

Let Let and and be given. We first prove that (1.1)-(1.4) with the initial condition

(1.6)

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 starts before the quantity becomes small because the predatory efficiency a₃ tends to zero, by evolution. We remark that although plays an important role in the non-evolutional system near equilibrium point, the quantity is crucial in the evolutional one. This is because the quantity 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 occur as in the case of non-evolutional model, while in other subdomain, evolution stabilizes large fluctuations of and. We also discuss the role of γ in (C.1), which is related with the sensitivity of evolution to the character bias. (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., decreases to zero.

2. 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

(2.1)

for some. Moreover we suppose that there exist and such that

(2.2)

The following local existence and uniqueness theorem is well known.

THEOREM 2.1. Assume (2.1) and (2.2). Then there exists a such that the system of Equations (1.1)- (1.4) with the initial conditions (1.6) has a unique continuously differentiable solution, in

In the following we study the existence of a global solution. We require the condition

(2.3)

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 (2.3) is satisfied. Then the system of Equations (1.1)-(1.4) with the initial condition (1.6) has a unique global solution in

Proof. First we will show the apriori estimate for all. Suppose that this is not true. Then, by the continuity of and in (2.3) we can take the smallest time such that Assume that If we set in (1.1)-(1.4), then we have

(2.4)

By the local existence and uniqueness theorem, Equations (2.4) with the initial condition has a unique solution. We denote the solution by. Then (1.1)-(1.4) with the initial value at has a solution By the uniqueness of the solution we obtain It follows that Because by (2.3), we have a contradiction. Hence we have By the continuity of one may assume that in a sufficiently small neighborhood of. Then, the second term in the right hand side of (1.1) satisfies in a sufficiently small neighborhood of On the other hand, since can be made arbitrarily small by taking a neighborhood of small, it follows that there. Hence is a decreasing function. This contradicts to Therefore, there is not such that which shows the desired estimate.

Next we will estimate N₂ from the above. Take that and add ε times (1.2) to (1.1). Then we have

(2.5)

where

Hence, by setting we obtain

(2.6)

Multiplying to both sides, and integrating from to we obtain

(2.7)

By the apriori estimate there exists M > 0 depending only on r, K and such that Hence we have

Because, we obtain

(2.8)

It follows that, for

(2.9)

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 Take ε so that and add ε times (1.3) to (1.2). Then we have

(2.10)

where

By setting we obtain the equation. Because this equation has a similar form as in the case, we can choose a constant depending only on and the initial values so that Then we argue in the same way and we obtain

(2.11)

In view of the definition of v we have

(2.12)

Next we will estimate from the below. By the estimates of and from the above there exists such that It follows that

By integrating from to t we obtain

(2.13)

We will estimate N₂ from the below. There exist constants depending on the equation and the initial values such that, Hence we have

By integrating the inequality from to t we obtain

(2.14)

The estimate of from the below can be shown by simple computations.

(2.15)

Next we will prove

(2.16)

Indeed, we have (2.16) for by the initial condition. It follows that if is sufficiently small, then (2.16) holds true.

In order to prove (2.16) we assume that there exists such that either or 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 from the above we have, for any, there exists a neighborhood V of such that if then and hold. Hence we have

(2.17)

If then the right hand side of (2.17) is negative. Therefore is decreasing near. This implies that does not tend to when. Because is continuous, we have. This is a contradiction. Hence we have the desired estimate.

We shall prove the existence of a global solution. Set and let be the maximal interval for which and are defined. If, then we are done. Assume that We will show that the limits and exist. We set where 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

(2.18)

By the apriori estimates from the above, is bounded on. Hence there exists M such that It follows that the limit exists. If we define, then is continuous up to. We will show that it is. For this purpose it is sufficient to show that We note that is Lipschitz continuous in each variable because we have apriori estimates of N and. 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 .We can solve (1.1)-(1.4) with the initial values and at. Then by the unique existence of the solution we can extend and to some neighborhood of. This contradicts to the definition of. Hence we have. This ends the proof.

Remark. 1) We remark that the apriori estimate of a solution does not depend on the cost of evolution 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., then (1.1)- (1.3) with the initial condition (1.6) has a unique global solution. Indeed, (1.1)-(1.4) can be split into (1.1)-(1.4), 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.

3. 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

(3.1)

Let be an integer and and be constants. We assume

(C.1)

We also assume that is twice continuously differentiable and nonnegative in the closed intervals and. If we denote the right-hand side of (3.1) by, then, by (C.1) we have

(3.2)

on We define on by the right-hand side of (3.2). Define, for

(3.3)

We first study the behavior of.

LEMMA 3.1. 1) Assume Suppose that. Then has a unique zero z = 0 in the interval and is negative on.

Assume. Then has simple zeros, and 0 on. The function is negative on the intervals and, while it is positive on and. (cf. Figure 1).

Moreover, there exists such that z₀ has an asymptotic behavior

(3.4)

when Similarly we have

(3.5)

when

2) Assume If then has simple zeros, and 0 on.

The function is negative on the intervals and while it is positive on and. (cf. Figure 1).

If then has a unique zero z = 0 in and is positive on (cf. Figure 2).

Proof. We divide the proof into 5 steps.

Step 1. By (C.1) we have

(3.6)

Set

(3.7)

and define

(3.8)

We consider the zeros and the sign of in the interval

Step 2. First we consider the case Assume that is an odd integer. Because is an even integer, we have We have and. Because we easily see that if, it follows that has no zero point on.

In order to study the zero of in (0,1), note. One easily see that the assumption is equivalent to Because on (0,1], we see that has only one zero point in the interval (0,1] if In view of (3.6) we conclude that has zero points and 0 in the interval for some. It is also clear that if the opposite inequality holds, then on.

Next we consider the case is even. By the same way as in the odd case, we have, and Since is strictly increasing on (0,1), there exists unique, such that In order to show that has no zero on we note

Because and

on, we see that has a unique zero point on. Clearly, has only one zero, because for all which proves the assertion. The sign of is almost clear from the definitions of and the argument in the above.

Step 3. We will show the asymptotic formula of in (3.4). In view of the argument in Step 2 we may consider

If we set, then we have

(3.9)

where is a polynomial of with positive coefficients. Hence we have

(3.10)

Hence, for sufficiently small we can uniquely solve (3.10). By an implicit function theorem we see that is a smooth function of such that It follows that

Therefore we have (3.4).

Step 4. Next we will prove (3.5). We will solve namely

(3.11)

for Hence we have

(3.12)

If does not converges to zero when then there exist c>0 and a sequence . Since the right-hand side of (3.12) is bounded for this leads to a contradiction. Hence is asymptotically equal to

By solving this relation we have

when By (3.11) we have . It follows that

By simple computations we obtain (3.5).

Step5. If then we have We can easily see that on The solution of is given by 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 We note that two functions are identical for For a small number, consider the case shown in Figure 2 for Then f looks like as in Figure 3 where new attractive equilibrium points appear near because we have made a modification to so that The new equilibrium point corresponds to that of with modulus larger than. 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 This fact is important since, if otherwise, the efficiency 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 with (cf. Figures 1 and 4).On the other hand, if, then the dynamics of f and in may be different, while in other part both are the same.

We also note that the apriori estimate holds for f. Therefore apart from the neighborhood of the dynamics of f is well approximated by that of, for which we can make concrete analysis of the dynamics, 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

(3.13)

(3.14)

(3.15)

(3.16)

It follows from (3.15) that

(3.17)

One may assume that A is small since is small. By (3.14) we have

(3.18)

Let us first consider the non-evolutional case, z = 0 or the case where evolution becomes stationary, namely. Then there exists C > 0 such that Hence is small and is close to K, by (3.13). It follows that there exists such that if is sufficiently small. In terms of (3.17) and (3.18) tends to infinity when. This implies a typical behavior of and around an equilibrium point when there is little effect of evolution. Numerical experiments show that the decrease of occurs soon after the orbit approaches to the equilibrium point, namely becomes sufficiently large.

Let us consider the evolutional case. Then the main difference from the non-evolutional case is that may tends to zero. For the sake of simplicity, let us consider the case or. Assume that there exists an orbit such that and grows large. Because also grows large, it follows from (3.4) that tends to zero, and we are in the situation that the orbit of tends to. Therefore tends to zero. Hence the boundedness of implies that becomes negative. Therefore, by (1.3), exponentially decreases. Note that the decrease of begins after exceeds a certain constant independent of and when. This exhibits a strong contrast to the non evolutional case where the collapse of occurs after becomes sufficiently large.

2) Effect of the parameter to evolution.

In the predatory efficiency in (C.1) represents sensitivity to z. Namely, as increases, for small z approaches to a constant function. The dynamics of z is quite different in the cases and. Indeed, if and

then evolution progresses. (cf. Figure 2 and Figure 3). The latter condition means either the cost of evolution is small, or. We note that the attracttive equilibrium points near have the effect to hold the orbits around. Conversely, if , then we see that fluctuations in progress and rest of evolution takes place.

In the case we have a different situation. Indeed, if then the evolution becomes stationary. If otherwise, then similar fluctuations in progress and rest of evolution as in the case takes place. We will show in the next section that in the linear case we have a sharp contrast to the case.

3) Fluctuations of and.

The rhythm of and is also observed in a nonevolutional system and it is related with the structure of a tea-cup attractor. We have a similar phenomenon for an evolutional system.

Let and be a solution of (1.1)- (1.4). One can show by Poincaré –Bendixon theorem that and 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 there exists such that if

then

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 and for some interval of time. Note that is small and the evolution becomes stationary, i.e.,.

In order to estimate T, we take, and. By integrating the equation of, one has

Hence, if we have

(3.19)

for sufficiently small, then we have from which we have the estimate of time length T, We have a similar condition like (3.19) in the general case by replacing 0 and T, respectively, by and A similar condition like (3.19) holds for some and T if we have an averaging property:

4) The limit case when evolution cost tends to zero.

We assume. If the evolution cost tends to zero, namely grows from zero to , then, by (3.4) and the definition of approaches to the origin.

Therefore, the evolution progresses, namely z approaches either of the points. It follows that the predatory efficiency. By the same reasoning as in 1) tends to zero. We note that in the limit case the third species dies out. This agrees with an ecological observation.

4. Evolution for a Linear Predatory Efficiency

We will discuss the evolution in the case

(4.1)

where is a real constant and .As in the previous case we make modifications of in some small neighborhood of the zero point such that.

For the sake of simplicity, we assume that. 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

We will study the dynamics of the evolution in relation with the populations and.We now define

(4.2)

The condition is equivalent to

By definition we may consider (4.3) in the set because, if otherwise, Set

and calculate the minimum of in I. It is taken at with the minimum value given by

We recall that is equivalent to Therefore, if modulo terms of namely

(4.4)

then there appear an attractive equilibrium point near the origin. This means that the predatory efficiency is close to a constant function if is sufficiently small. Indeed, the equilibrium point can be estimated as

In view of the linearity of and the smallness of near the equilibrium point we see that is almost constant for small changes of.

Suppose now that (4.4) does not hold. Then the attracttive equilibrium point near disappears, and there remains an attractive equilibrium point near the zero of.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.

5. 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 and be the solution of (1.1)-(1.4). Let be the solution of the non-evolutional system (1.1)-(1.3), namely Then we have

THEOREM 5.1. Assume (2.3). Let be arbitrarily given. Then we have

uniformly in t on

Proof. By integrating (1.4) we have

(5.1)

If we make the change of variables, , then we have

(5.2)

where and

Because is uniformly bounded in by the apriori estimate, it follows that times the integrand is uniformly bounded in when. Hence, the modulus of the integral can be bounded by a constant times It follows that

uniformly in when. This entails that uniformly in.

For the sake of simplicity we write (1.1)-(1.3) in

where we use the same notation F as in (2.18). Since

we have

(5.3)

where the absolute value of a vector means the norm of a vector. Because we have the uniform estimate of in by (1) of Remark in Section 2, we have

for some independent of.

By (1.5) and (5.2) we have

(5.4)

Because there exists a constant independent of such that the right hand side of (5.4) can be estimated by It follows that for any there exist and A>0 such that, for

Therefore we have

By Gronwall’s inequality we obtain, for

Because is arbitrary, we have the desired estimate.

6. Discussion

Evolutional Lotka-Volterra system does not seem to be well understood analytically except for the case of two species. In this paper, we have studied how the evolutional change of a character influences global behaviors of a Lotka-Volterra system for three species. We introduced an evolutional equation based on a quantitative genetic model into a Lotka-Volterra system of equations and we proved the existence and the uniqueness of a global solution as well as apriori estimates of a solution. By virtue of these properties, we have given analytical proofs of properties which are different from the nonevolutional system. We hope that some of the properties shown in this paper hold for more general food web settings. It is also interesting to make numerical analysis of our theory in order to understand the effect of evolution. The study of these problems will be left for the future study.

REFERENCES

Appendix

(A) The following lemma is used in Section 5.

LEMMA. (Gronwall) Let be a closed interval and let Let u be continuously differentiable in I such that, for some constants and the inequality

holds true for. Then we have on I.

Proof. For the sake of simplicity, we consider the case Denoting the right hand side of the inequality by v(t) we have the relations, and

Multiplying the last inequality with, we have By integration we get

(B) We will briefly show how to deduce (1.4) from the theory of quantitative genetics. Let, 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 We assume that (cf. [3-5])

where is the cost of evolution. By definition we have

(7.1)

Here, and are certain constants. We have

Because one may regards as a constant function when z varies, the right-hand side of (7.1) can be replaced by

Hence we obtain (1.4).