a nonautonomous predator-prey system arising from...

A Nonautonomous Predator-Prey SystemArising from Coagulation TheoryFERNANDO P. DA COSTA1, 2 & JOA

~O T. PINTO1, 3

ABSTRACT: A recent investigation of Budác et al., on the selfsimilar behaviour of solutions to a model of coagulationwith maximum size [Oxford Center for Nonlinear PE, Report no. OxPDE-10/01, June 2010] led us to consider arelated nonautonomous Lotka-Volterra predator-prey system in which the vector field of the predator equationconverges to zero as t +. The solutions of the system show a behaviour distinct from those of either autonomousor periodic analogs. A partial numerical and analytical study of these systems is initiated. An ecological interpretationof this type of systems is proposed.

AMS MATHEMATICS SUBJECT CLASSIFICATION: 34A34, 34C11, 34C60, 92D25.

International Journal of Biomathematics and Biostatistics,Vol. 1, No. 2, July-December 2010, pp. 129-140

Fernando P. da Costa: 1 Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, TU Lisbon, Av.Rovisco Pais 1, 1049-001 Lisboa, Portugal.2 Departamento de Ciências e Tecnologia, Universidade Aberta, Rua da Escola Politécnica 141-147, 1269-001 Lisboa, Portugal.

Joao T. Pinto: 1 Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, TU Lisbon, Av. Rovisco Pais1, 1049-001 Lisboa, Portugal.3 Departamento de Matemática, Instituto Superior Técnico, TU Lisbon, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal.

1. INTRODUCTION

In a recent mathematical study of elastocapillary coalescence, [1], a coagulation problem has been proposedin which the coagulation between two particles of sizes x and y occurs only if the sum x + y of their sizes isexactly equal to a given increasing function M () of the time .

By normalizing the total mass to 1 and interpreting M () as a rescaled time, the coagulation problemcan be written as follows [2]:

0( , ) ( , ) ( , )( )

kc

x c x c xN

, (1.1)

0( ) ( , )

N c x dx , (1.2)

0( , ) 1

xc x dx , (1.3)

where k0 is a positive constant, and c (, x) denotes the concentration of clusters with size x ∈ [0, ] at time

1.

Global existence and uniqueness of (1.1)-(1.3) has been established in [2, Section 2]. In [2, Section 3]the authors start the study of self-similar solutions. These are solutions to (1.1)-(1.3) that have the form

I BBSerials Publications

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130 INTERNATIONAL JOURNAL OF BIOMATHEMATICS AND BIOSTATISTICS (IJBB)

( ) , if ,( , )

0, if ,

xG x

c x

x

(1.4)

where G : [0, 1] +0 is a C1 function.

Substituting (1.4) into (1.1)-(1.3) and taking into consideration some scale invariances of the resultingequation one concludes that G must be a solution to the equation

yG(y) = G (y) (G (1 – y) – 2), y [0, 1]. (1.5)

In [2, Lemma 3.1] it has been proved that, for all given G (1/2) = > 0, there exists a unique positivesolution G of (1.5) in (0, 1). It is fairly easy to recognize just by inspection of (1.5) that G (y) 2 is the onlyconstant positive solution to (1.5).

In [2] the authors also present a brief numerical study of the solutions to (1.5) in which they identifytwo different types of solutions, that they denote as subcritical and supercritical, and that are distinguishedby the fact that G (1/2) = < 2, and G (1/2) = > 2, respectively. All solutions of a given type have the samequalitative behaviour, which is shown in Fig. 1. Their main qualitative features are the following:

Figure 1: Several Subcritical (Left) and Supercritical (Right) Solutions G of (1.5), Reproduced from [2]. For the Subcritical Cases,Note the Existence of Two Local Maxima, Located Very Close to the Boundary of the Interval (0, 1)

Supercritical solutions have a unique local maximum, located at a point yM > 1

2

Subcritical solutions have a unique local minimum, located at a point ym > 1

2 , and have two localmaxima, located very close to each of the boundary points of (0, 1).

We had two original goals: (i) to understand the different qualitative behaviour of the subcritical andthe supercritical solutions, and (ii) to rigorously justify the numerical evidence presented in [2]. Theunderstanding of the qualitatively different behaviour of these two types of solutions is fairly easy to achievenumerically by a change of variables that transforms (1.5) with the condition G (1/2) = > 0 into anonautonomous Lotka-Volterra predator-prey system, although the rigorous proofs of the numerical behaviourexhibited by this system still eludes us at present. However, the behaviour of solutions to the nonautonomousLotka-Volterra system, and to other related ones, turned out to be both intriguing and challenging, and so,in this note we intend mainly to draw the attention of the reader to these classes of nonautonomous systemsthat, so far, seem to have received scant attention in the literature.

2. INSIGHT PROVIDED BY A NONAUTONOMOUS PREDATOR-PREY SYSTEM

Being interested in the study of the positive solutions to the problem

A NONAUTONOMOUS PREDATOR-PREY SYSTEM ARISING FROM COAGULATION THEORY 131

yG(y) = G (y) (G (1 – y) – 2), y (0, 1) (2.1)

G (1/2) = > 0, (2.2)

we define (u~, v~) : (0, 1) 2 by

u~ (y) := G (y), (2.3)

v~ (y) := G (1 – y). (2.4)

With this notation equation (2.1) reads as yµ~ (y) = u~ (y) (v~ (y) – 2). Furthermore, differentiating v~ andusing (2.1) we immediately obtain the following differential equation (1 – y) v~ (y) = – v~ (y) (u~ (y) – 2). In thenew variables, condition (2.2) translates to u~ (1/2) = v~ (1/2) = α > 0. Thus, we conclude that G is a solutionof (2.1)-(2.2) if and only if (u~, v~) is a solution of the system

1( ) ( ) ( ( ) 2)

1( ) ( ) ( ( ) 2),

1

u y u y v yy

v y v y u yy

(2.5)

for y (0, 1), and initial condition (u~, v~)(1/2) = (, ).

Observe that this system is invariant for the symmetry (y, u~(y), v~(y)) (1 – y, v~(y), u~(y)) and so, forsolutions with the symmetric initial condition we are presently considering, we have u~(1 – y) = v~(y). Thisimplies that we need only to study the solutions of (2.5) for y either in [ 1

2 , 1) or in (0, 12 ], and read the

behaviour in the other interval by changing the roles of u~ and v~. So, from this point onwards we willconsider (2.5) in [ 1

2 , 1).

Consider a new independent variable t such that 11 y dy dt and t = 0 when y = 1/2. Upon

integration, the relation t (·) : [ 12 , 1) [0,) between t and y, is given by t = t (y) = – log (2 (1 – y)). Let

u (t) := u~ (y (t))

v (t) := v~ (y (t)),

where t y (t) is the inverse function of the function y t (y) given above. With these new variables ourproblem becomes

( ) ( 2), 0,

(2 )

u g t u vt

v v u

(2.6)

with initial condition (u, v) (0) = (, ), where the dot denotes the derivative with respect to the variable t

and g (t) := 12 1te

. In Fig. 2 we present the projections onto the (u, v) space of the graphs of some numericallycomputed solutions of this initial value problem.

In Fig. 3 we present an example of a single orbit of a subcritical solution and another one of a supercriticalsolution. These examples allow us to understand the different qualitative behaviour exhibited by sub andsupercritical solutions and shown in Fig. 1. Let us see how.

Consider first the subcritical solution in the left of Fig. 3. Consider the u-component of that solution: itstarts by decreasing until attaining a minimum (at the point where the orbit crosses the straight line v = 2).That occurs at some time t

m > 0 to which corresponds a value y

m > 1

2 for which G (·) attains its minimumvalue G (y

m) = u (t

m). For t > t

m the u component of the orbit starts increasing until it attains a (barely

perceptible) maximum when the orbit again crosses v = 2, after which u decreases very slightly while the


orbit approches the u-axis. This maximum of u corresponds to the maximum of G that is attained very closeto the point y = 1 and that is also barely discernible in the plots shown in Fig. 1.

If we now consider the behaviour of the v-component of the same solution, starting from the initialpoint (, ), we know that, as explained above, it has a direct relation with the behaviour of G for y < 1

2 .From the left of Fig. 3, we see that the v-component of the solution increases monotonically until it attainsa maximum (when the orbit crosses the line u = 2), after which it decays monotonically to zero. Thus, weconclude that G (y) increases while y decreases from 1

2 to a certain value yM (0, 1

2 ), from which it thendecreases to zero as y 0 (this decreasing to zero is not really visible in the plots presented in Fig. 1).

The use of the plot of the supercritical orbit in the right of Fig. 3 to provide an explanation for thebehaviour of the supercritical solutions in Fig. 1 is entirely analogous to the previous case an is left to thereader.

In the remainder of this paper we will present some analytical results and numerical evidence concerningthe behaviour of solutions to (2.6) and related nonautonomous Lotka-Volterra systems.

Figure 2: Projection onto the (u, v) Space of Some Solutions to (2.6) with Initial Data (u, v) (0) = (, )

Figure 3: Projection onto the (u, v) Space of a Subcritical and a Supercritical Solution to (2.6)(Left and Right Graphics, Respectively)


3. GENERAL ANALYTICAL RESULTS

Consider the class of systems

( ) ( 2), 0,

(2 )

u g t u v

tv v u

(3.1)

where g, > 0, satisfies the assumption

(H) g : [0, +) (0, 1] is a continuous function satisfying g(0) = 1, g(t) < 1, for t > 0, and g(t)et

is bounded and bounded away from zero as t +, i.e., g(t)e t [K

0, K

1] holds true for all t 0

for some constants K1 K

0 > 0 independent of t.

Clearly, the function g in (2.6) satisfies (H) with = 1.

We are only interested in non-negative solutions to (3.1), so we start by establishing the invariance ofthe closure of + := + +, the positive cone of 2.

Proposition 1: The sets +, +, and {(2, 2)}, are (positively) invariant for the evolution generatedby (3.1).

Proof: With the continuity of g assumed in (H), system (3.1) satisfies the conditions of the Picard-Lindelöf theorem. Thus, solutions to initial value problems for (3.1) are unique. For the coordinate axis,+ {0} and {0}+, the invariance result is obvious since, in the first case, (3.1) reduces to u• = – 2gu,v• = 0, and so the solution stays in the u-axis, and in the second case, (3.1) reduces to u• = 0, v• = 2v and thesolution also stays in the v-axis. By the uniqueness of solutions and the form of the system in + nosolution starting in + can converge to a point in + in finite time, and so it must remain in +. Theinvariance of {(2, 2)} is also trivial by the uniqueness of solutions.

Remember that in the case when the function g is constant (i.e., for autonomous predator-prey systems)the auxiliary function : + defined by (u, v) = (u – 2 log u) + (v – 2 log v) is a first integral for thesystem. For nonconstant g we have

( , ) (1 ( )) ( 2) ( 2) ,u v g t u v

and the behaviour of along solutions depends of where in + the solution happens to be at a given time t.

For notational convenience we shall decompose + into the lines u = 2 and v = 2 and the following fourdisjoint open sets:

1 := {(u, v) : u > 2, 0 < v < 2}

2 := {(u, v) : 0 < u < 2, 0 < v < 2}

3 := {(u, v) : 0 < u < 2, v > 2}

4 := {(u, v) : u > 2, v > 2}

Note that, in j, the value of increases along solutions if j is odd, and decrease if j is even. However,

the behaviour of along solutions is really not used in the study presented next.

The assumption that g is bounded above by 1 implies that the signs of u and v are constant in each j,

and, from the right hand side of (3.1), the following holds:


1

2

3

4

( 1, 1), if ( , )

( 1, 1), if ( , )( , )

( 1, 1), if ( , )

( 1, 1), if ( , ) ,

u v

u vsgn v

u v

u v

(3.2)

We now start the study of the possible limit points of solutions to (3.1).

Proposition 2: Let (u, v) be a solution of (3.1) with initial datum in +. If the solution is bounded andremains in a single

j after a sufficiently long time, then it must converge to a point (u, v) in := {u = 2,

v 0} {u 2, v = 0}.

Proof: We start by noting that the j in the statement can only be

1 or

3. In fact, suppose t

0 is a

sufficiently large time after which the solution forever stays in, say, 2. Let (u

0, v

0) = (u (t

0), v (t

0))

2.

Since, by assumption, the solution stays in 2 for all t t

0, and since u• < 0 in this region, we have

u (t) < u0 < 2, which implies that v must satisfy v• = v (2 – u) > (2 – u

0) v. Thus v (t) v

0e(2 – u0)(t – t0) > 2 if we

choose t sufficiently large. But this means that (u, v) is no longer in 2, a contradiction.

A similar argument applies under the assumption that the solution stays in 4 for all sufficiently large

times. Since u• > 0 for solutions in 4 we have that u (t) > u

0 > 2 for all t > t

0 and thus v• < (2 – u

0) v. But then

v (t) 0 as t, a contradiction with the assumption that (u, v) will stay in 4.

Actually, the previous arguments imply that solutions in 2 will eventually enter

3, and, analogously,

solutions in 4 will eventually enter

1.

Now suppose a solution is in 1 at time t

0 and remains there for all larger times. Since u• < 0 in

1 we

have that u (t) is strictly decreasing and bounded below by 2 (since the orbit remains in 1). So, u (t) must

converge to some u 2. If u = 2 the limit point is in {u = 2, 0 v < 2} . Otherwise, we haveu (t) > u > 2 from which we conclude that v (t) will satisfy v• < (2 – u) v and thus converges exponentiallyto zero as t. Hence (u, v) converges to a point in {u > 2, v = 0} .

To conclude, assume the solution is in 3 at time t

0 and remains there for all larger times. The argument

is analogous to the previous one: since u• > 0 in 3, the component u () of the solution is strictly increasing

and bounded above by 2 (since the orbit remains in 3, by assumption). Thus u (t) must converge to some

u 2. If u < 2, then the equation for v would imply that ν• (2 – u) ν which contradicts the boundednessassumption about the solution. Thus we must have u = 2 and v (t) bounded (by assumption) and monotoneincreasing (because the orbit remains in

3). Hence (u, v) must converge to some point in {u = 2, v > 2},

and this completes the proof.

It is worth noting that the boundedness assumption in Proposition 2 was only used in the study of thebehaviour of solutions remaining in

3: solutions cannot remain in

2 and

4, and those remaining in

1

are automatically bounded. In the next result we relax the boundedness condition.

Proposition 3: Let (u, v) be a solution of (3.1) that remains in 3 for all sufficiently large times. Then,

either (u, v) is bounded (and converges to a point in {u = 2, v > 2}) or (u, v) (u,) when t, forsome u [2 – , 2].

Proof: The bounded case was considered in the proof of Proposition 2. Let (u, v) be an unboundedsolution of (3.1) that stays in

3 for all sufficiently large time. From (3.2) u is monotonically increasing

and so it converges to a value u 2 and u (t) < u for all t > t0. Let us assume that u < 2 – . Define

:= 2 – – u > 0. The upper bound on u implies that v• ( + )v and thus v (t) v0e( + ) (t – t0). Plugging this

bound for v, and the lower bound for g given in (H), into the equation for u we obtain, after integrating intime from t

0 to an arbitrary t > t

0,


0 00 0( ) ( )0 0 0

0

2( )log ( 1) (1 )

t tt t t tK v e K eu t

e eu

. (3.3)

Now, since > 0, the first term in the right hand side of (3.3) diverges to infinity as t and so u isunbounded, which contradicts the assumption that (u, v) remains in

3, and concludes the proof.

In Fig. 4 we present some orbits of solutions to (3.1) that illustrate the distinct behaviours discussed inPropositions 2 and 3.

To complete the general picture concerning the long time behaviour of solutions to (3.1) it remains toconsider the possibility that solutions do not end up in one of the

j after sufficiently long time, but instead

visit each j infinitely often (clearly, by (3.2), if a solutions does not stabilize in a single

j it must visit all

of them infinitely often). Observe that in the autonomous case (when g 1, say) this is exactly whathappens, as can be easily concluded from the fact that is a first integral of the system and so solutions lieon the level sets of , which are closed curves.

The examples of orbits presented in Figs. 2 and 4 are, in a sense, misleading, since for other choices ofg or initial conditions we can produce orbits that visit each of the sets

j a large number of times, as shown

in the example in Fig. 5. However, as we shall prove in Proposition 4, with assumption (H) the number oftimes an orbit visits each

j is always necessarily finite, and so the general situation is the one described by

Propositions 2 and 3.

Figure 4: Examples of Orbits of Solutions to (3.1) Exhibiting the Behaviours Discussed in Propositions 2 and 3. The Function g is

g(t) = 12 1te

Considered in System (2.6). Note the existence of unbounded orbits, as well as orbits converging to points in

. In (a): Initial Data are Points Along the Line v – 2 = (u – 2) tan (/8) with Distances to (2, 2) Equal to 0.2, 0.4, ..., 0.8,1.01046, 1.2, 1.4, ..., 2.8, 3.0, all with u > 2. In (b): Initial Data are Points Along the Line v – 2 = (u – 2) tan (3/8) withDistances to (2, 2) Equal to 0.2, 0.4, ..., 1.2, 1.2926, 1.4, 1.6, 1.8, 2.0, all with u < 2. In Both Cases the Initial Time is t

0 = 0.

Note That the Scales of Coordinate Axis in (a) and in (b) are Different

The proof of this result is based on a change to polar coordinates relative to the sets j (i.e., the set

j is

essentially the j th-quadrant in the new variables) and the use of a nonlinear variation of parameters formulain the analysis of the equation for the angle variable.

The approach requires an auxiliary result about the behaviour of the orbits that repeatedly visit all thesets

j. That behaviour is apparent in the example shown in Fig. 5: orbits circling the equilibrium point

(2, 2) are being compressed along the line u = 2. The following Lemma states the needed result.

Lemma 1: Let (u, v) be a solution to (3.1) with initial data in +. Suppose that u (tn) 2 for a sequence

tn. Then, for each arbitrary > 0, there is > 0 such that, for all t , u (t) 2 – . Suppose instead


that u (tn) 2 for a sequence t

n. Then, for arbitrary > 0, there is > 0 such that u (t) 2 + , for all

t . If, in particular, both such sequences (tn) and (t

n) exist, then u (t) 2, as t.

Proof: By the invariance of + we have, for all t 0,

( )( )( ( ) 2) 2 ( )

( ) u t

g t v t g tu t

,

But, by the upper bound on g in (H), we have, if 0 t0 t,

0

0

1

00

( ) 2log 2 ( ) 2 ( )

( )

t t

t t

u t Kg s ds g s ds e

u t

.

Since t0 1 02K te is strictly increasing and converges to 0, as t

0 +, we conclude that, given a

small > 0, if t0 is sufficiently large then u (t) u (t

0)(1 – /2), for all t t

0. This clearly implies the first

assertion.

For the second assertion, first observe that if for all t large enough u (t) < 2, then the conclusion istrivial. Hence we assume that for every t there exists a t > t such that u ( t ) 2. Since we can now apply ourfirst assertion, for a small > 0, there is a t

0 0 such that (u (t

0), v (t

0))

2, and furthermore, for all t t

0,

u (t) 2 – + . Then, for all t t0,

( )2 ( )

( )

v tu t

v t

,

and therefore v (t) v (t0) e( – ) (t – t0), and by the u•-equation,

0

0

( ) ( )0

0

( )log ( ) ( ( ) 2)

( )

t s t

t

u tg s v t e ds

u t

.

Again by the upper bound on g in the hypothesis (H), and since v (t0) 2,

0 0 0

0

( ) ( ) ( ) 11

00

( ) 2log 2 ( ) 2

( )

t s t t ts

t t

u t Kg s e ds K e e ds e

u t.

Figure 5: Example of an Orbit of (3.1) Visiting Several Times Each of thej (Before Eventually Staying Forever in

1). The Initial

Condition is Subcritical (u0, v

0, t

0) = (1.6, 1.6, 0) and is Denoted by the Black Dot. The Nonautonomous Function is

g0.05

(t) = (2e0.05t – 1)–1


Hence, for every > 0, if t0 is chosen large enough then u(t) u (t

0)(1 + /2) thus implying the second

assertion.

Proposition 4: Let (u, v) be an arbitrary solution of (3.1) in +. Then, (u, v) cannot visit any of the sets

j infinitely often and hence the cases in Propositions 2 and 3 are the only possible ones.

Proof: We introduce polar coordinates r, centered at (2, 2) with increasing clockwisely starting with = 0 at {(u, 2) : u > 0}. Then

u – 2 = r cos , v – 2 = – r sin ,

and the differential equation for obtained from (3.1) can be written in the form

2( ) ( ) ( ) cosg t u t h t . (3.4)

where

( ) ( ) ( ) ( )h t v t g t u t .

Our goal is to prove that () is bounded in [0,). We will suppose that this is false (and thus that thesolution visits each of the

j infinitely often) and shall seek a contradiction. We use the representation of

the solutions of (3.4) given by a nonlinear version of the variation of parameters method (see [4], forinstance) to arrive at a contradiction. Consider a nonconstant solution (u (t), v (t)) in +, defined for t 0.For arbitrary t, t

0 0 and real

0 let us write the solution of (3.4) satisfying (t

0) =

0 as (t, t

0,

0). Obviously

the meaningful solution is (, 0, 0) when

0 is the angle in (–, ] that corresponds to (u (0), v (0)). Also,

for every t, t0 0 and every real

0, let us denote by (t, t

0,

0) the solution of

20 0( ) cos , ( )h t t (3.5)

This is a separable ODE that can be readily solved. Writing

0

00 0

1( , ) : ( ) , ( ) :

2

t

tH t t h s ds K

where [x] := max {k : k x}, and

0 0 0

0 0

0

arctan (tan ( , )), if2( , , ) :

, if2 2

H t tt t

then, for every (t0,

0) +

0 , the solution of (3.5) is given by

(t, t0,

0) = K (

0) + (t, t

0,

0), (3.6)

By basic ODE theory, the map : (+0 )2 is smooth. Then, the variation of parameters method

consists in finding, for given (t0,

0) +

0 , a function (t) such that (t0) =

0, and furthermore

(t, t0,

0) = (t, t

0, (t)), (3.7)

for all t 0 in the maximal interval of existence of (, t0,

0). Substituting (3.7) into (3.4) and using (3.5),

one readily concludes that (t) must be the solution of the Cauchy problem,

0

10 0

0 00 ( )

( , , )( ) ( ) ( ), ( )

t

t tt g t u t t

(3.8)


In our case,

0

10 0 2 2

0 00 ( )

( , , )1 ( , ) sin 2 ( ) ( , ) cos ( )

t

t tH t t t H t t t

.

We now prove that, assuming the solution (u, v) of (3.1) visits each j infinitely often entails the

boundeness of (t) for t 0, which, by the nonlinear variation of parameters formula (3.7)-(3.8), impliesthe boundedness of (), an obvious contradiction.

From (3.8), for t t0 we have

0

20 00( ) (1 ( , ) ) ( ) ( )( , )

t

tt H s t g s u s dsH s t . (3.9)

Assuming our absurd hypothesis, it follows from Lemma 1 that u (t) 2, as t +. Therefore, given (0, /2) there is t

0 0 sufficiently large so that, for all t t

0, | u (t) – 2 | < and, by the v•-equation

| h (t) | v (t) + g(t) u(t) v (t0) e(t – t0) + (2 + ) K

1e– t

which in turns implies

0( )0 10

( ) (2 )( , )

t tv t KeH t t ,

and therefore, by (H),

( )1 0 10

( ) (2 )( )( , )

t tK v t K

g t e eH t t ,

2 32 22 ( 2 ) ( )1 0 1 0 1

0 2 2

(2 )( ) 2(2 ) ( )( , ) ( )

t t tKK v t K v t

H t t g t e e e .

But then, the integrals0

( )t

g s ds

,0

0 ( )( , )

tg s dsH s t and

0

20( , ) ( )

tH s t g s ds

are all finite.Thus, by (3.9), (t) is bounded for t t

0 and so also for t 0. This implies the boundedness of t K ((t))

in [0, +) which, through (3.6) and (3.7), would contradict the hypothesis of the unboundedness oft (t, 0,

0) for t 0. Therefore, we conclude that, for every nonconstant solution (u (), v ()) in +, the

corresponding polar angle function () is bounded for t 0.

4. FINAL REMARKS

4.1 On the Numerical Evidence

At this point we would like to call the reader’s attention to the initial goal of the present note: we wanted tounderstand and prove the behaviour of the subcritical and supercritical solutions of (1.5) shown in Fig. 1.This is equivalent to prove that solutions to the Lotka-Volterra predator-prey system (2.6) with initialconditions in the diagonal v = u = have the behaviour shown in Fig. 2 or, equivalently, that for allt [0, +), the angle function () is monotone increasing and satisfies (t) [– /4, /2) if > 2, and (t) [– 5/4, /2) if < 2.

In the last section we proved a result that is, on one hand, somewhat weaker, but, on the other hand, isvalid for more general functions g and initial data: the boundedness of () which implies that, underassumption (H), solutions to (3.1) do not circle around the equilibrium point (2, 2) indefinitely. Furthermore,the numerical results illustrated in Figs. 4 and 5 suggest that what was shown in Fig. 2 concerning the


behaviour of all sub- and supercritical solutions is a rare event that is not to be expected for more general(non-diagonal) initial data or functions g.

Heuristically, we can understand the behaviour exhibited by the solutions of (3.1) having in mind thebehaviour that would correspond to a constant function g. For small values of t (how small depends onhow fast g(t) decays to zero) the system will aproximately behave as if g were a positive constant: theorbits tend to circle the equilibrium point (2, 2). While t is such that g(t) remains, in some sense, appreciable,the orbit will essentially continue to circle round (2, 2) but, for large enough t (and, again, how largedepends on the decay rate ), g(t) 0 and so the orbits tend to align with the orbits of the degeneratesystem with g = 0, which are vertical orbits parallel to u = 2. It is a numerical evidence that for nonautonomousfunctions in (3.1) with close to 1 there is a clear transition between “small” and “large” values of t (in thesense mentioned above). A graphical example of this evidence is presented in Fig. 6, where that transitionis marked by the curve * that corresponds to the isochronic set1 (T ) for T = 7.7, a value of time for whichg(7.7) = (2e7.7 – 1)–1 2 10– 4. Initial data on different manifolds will result in distinct curves * and whatis observed numerically is that for initial conditions in the diagonal v = u = < 2 the transition correspondto a curve that lies entirely inside

4, and so all the orbits with the initial data considered in Fig. 6 would

behave like those converging to the u-axis. A similar behaviour occurs for initial data with u0 > 2, such as

was presented in Fig. 4(a), and so we shall not further elaborate this point here.

Figure 6: Enlargement of Fig. 4(b) in Order to Make Clear the Points of the Orbits where the Distinction between “Small” and“Large” Times Take Place. The Line *, Corresponding to Points of the Orbits Attained at t 7.7, is in the Border Regionbetween “Small” and “Large” Times, for These Particular g and Initial Data

The study of the geometry of the isochronic sets (t) and its t-evolution is an important tool is theanalysis of the qualitative behaviour of solutions (see an example of this approach in [3], for a differentkind of system) and one can expect that a careful study of the geometry of (t) will also be useful in thiscase. Up to the present time, we could not yet successfully implement this approach.

4.2 On a Population Dynamics Interpretation

A final note concerns a possible population dynamics interpretation of the Lotka-Volterra system (3.1).System (3.1) is, formally, a predator-prey system for a population of predators with concentration u, andpreys with concentration v, uniformly mixed in a fixed isolated spatial region. However, in the biomathematicsliterature the vector fields considered in predatory-prey systems are either time independent or, if time


dependent, the time dependent coefficients are bounded and bounded away from zero, and in many caseseven periodicity is assumed. To the best of our knowledge, predatory-prey systems in which one of thecomponents of the vector field is decaying to zero as t + has not been considered previously in thebiomathematics literature. So, it is a challeging non-mathematical problem to devise an ecological situationto which a model such as (3.1) could apply, at least as a crude model. We think that such a situation isprovided by the domestication process of a population of wild predators, as we shall now describe. Let uscall the predators (wild) cats and the preys mice.

Assume that into the population of cats and mice, living freely in nature, a human starts artificiallyfeeding the cats (the intensity of the artificial feeding process is measure by the parameter > 0). The effectof this artificial feeding of the cats in the system is primarily felt in the cats’ dynamics: the rate of change ofthe number of cats, u• , starts to become indifferent to the hunting activity, since most of the cats’ foodnecessities are progressively being satisfied by the human feeder (mathematically this is translated in thefact that the vector field in the equation for u• is decaying to zero). However, the wild nature of the cats arenot quickly changed by their artificial feeding: the hunting instinct still operates and they go on huntingmice, essentially for fun2. So the hunting has less and less effect on the cats’ population rate of change, butit is felt by the mice as before, i.e., independently of the human feeding of the cats. Now, the final (i.e., ast +) population level of the cats will only depend on the initial population level and on intensity of theartificial feeding. If the final population level is larger than the equilibrium level (in our example thismeans larger than 2) then the mice population is subject to overhunting and becomes extinct. On the otherhand, if the final population level of the cats is smaller than the equilibrium level, the effect of the huntingis unable to prevent the explosive growth of the mice population. However, note that due to the artificialfeeding of the cats their population is insensitive to either the extinction or the explosion of the micepopulation.

Clearly, the use of nonautonomous systems analogous to (3.1) to model this ecological situation with aminimum of reasonableness will require the introduction of some modifications in the model, for instancethe inclusion of self-inhibition terms in the prey and predator populations comes imediately to mind. Itseems to us that the resulting classes of predator-prey systems are worth exploring.

ACKNOWLEDMENTS

We thank the authors of [2] for the permition to use Figure 1. This work was done with partial finacialsupport of FCT (Portugal)

NOTES

1. The isochronic set is the set (t) of points (u, v) + attained in t units of time by the solutions of the system with initialdata in a given fixed manifold in + {0}.

2. This behaviour can be seen in present day domestic cats, that still like to “play around” with mice, sometimes killing themin the process, but generally not eating them.

REFERENCES

[1] A. Boudaoud, J. Bico, and B. Roman, (2007), Elastocapillary Coalescence: Aggregation and Fragmentation with a MaximalSize, Phys. Rev. E, 76, 060102(R).

[2] O. Budác, M. Herrmann, B. Niethammer, and A. Spielmann, (2010), A Toy Model for Mass Aggregation with MaximalSize, Oxford Center for Nonlinear PE, Report no. OxPDE-10/01, 13, (June).

[3] F. P. da Costa, M. Grinfeld, N. J. Mottram, and J. T. Pinto, (2009), Uniqueness in the Freedericksz Transition with WeakAnchoring, J. Diff. Equations, 246, 2590-2600.

[4] M. E. Lord, and A. R. Mitchell, (1978), A New Approach to the Method of Nonlinear Variation of Parameters, Appl. Math.Comput., 4(2), 95-105.

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