Extinction and coexistence of predators Текст научной статьи по специальности «Математика»

MSC 2010: 34C05, 92D25

Extinction and coexistence of predators

A. V. Osipov*, G. J. Soderbacka**

* Saint Petersburg University,

Staryi Peterhof 198504, RUSSIA.E-mail: av_osipov@mail.ru ** Abo Akademi, Abo,

Turku 20500, FINLAND. E-mail: gsoderba@abo.fi

Abstracts. We study conditions for extinction and coexistence of predators in a general family of systems with many predators feeding on the same prey. We call the system degenerated in the case some predators go extinct and the essential dynamics occurs in a lower dimensional subspace. Keywords: predator, prey, extinction, dissipativity, periodic solution.

1. Introduction

We consider the (n + 1)-dimensional system

n

Xi = &i(s)xi, s = h(s) ^i(s)xi, i = 1, 2,..., n. (1.1)

i=i

Ai : All the considered functions are of the class C2[0, to) and the variables xi and s are non-negative: xi > 0, s > 0.

A2 : fa(0) = 0, ^i(s) > 0 for s > 0.

Here and further we will suppose that i takes values from the set {1, 2,... ,n}. A3 : $i(s) > 0 for s > 0 and there exists Xi > 0 such that <pi(\i) = 0.

A4 : h(0) = h(1) = 0, h'(1) < 0 and h''(s) < 0 for s > 0.

A5 : 0 < Xn < ••• < X2 < Xi < 1.

These and analogous systems for different assumptions on the right hand sides have been considered by many authors [1]-[5] of both mathematical and application view point. We note that our reference list cannot be full. It should be too large.

Analogous systems are considered from another point of view in [6, 7]. However the questions of dissipativity and existing of inner set are on the first place here also.

Systems with the following special choice of functions

s

h(s) = ys(K - s), ^i(s) = ai—--, <i(s) = mi^i(s) - di, (1.2)

s + ai

where all introduced parameters are positive, have been examined more extensively.

© A. V. OSIPOV, G. S. SODERBACKA

It is important to say, that this special choice of model is interesting from the mathematical point of view even in the three dimensional case. For instance, in [8, 9] the questions, connected with the problem of the limit cycle uniqueness in the coordinate planes are considered. This problem is connected with the stability of the planes and, consequently, with the problem of the inner set existing.

Such systems can have chaotic regimes. They were studied in detail in [11, 12]. Particularly bifurcation diagrams with a chain of period doubling bifurcations for system (1.1) with (1.2)were given. The functions

h(s) = s(1 - s),

Ms)

s - Xi

A(s)

s + ai

:i.3)

were chosen as an example. In this paper system 1.1 with these functions will be called standard system. We consider a broad class of systems of type (1.1), including right hand sides of type (1.2) and standard system as main example.

The conditions assumed for the system are divided into two groups. In the first group we have assumptions of general type (Ai-A5). In the other group we have assumptions of technical type arising in the formulation of statements (D6-D7 and A6-A7). It is easy to verify that all standard systems are included.

It follows from A1-A5 that the system has n +1 equilibria:

0(0,..., 0, 0) which is a saddle with n-dimensional stable manifold s = 0 and one dimensional unstable manifold: xi = 0(i = 1,... ,n), 0 <s< 1.

0'(0,..., 0,1) which is a saddle with one dimensional stable manifold xi = 0 (i = n) and n-dimensional unstable manifold, which we will denote by Wu(0').

h(Xi)

1

One point in each hyperplane: 0i: s The Jacobian matrix has the form

Xi, Xj = 0, j = i,

Xi

^i(Xi)'

Ф1 0 .. .. Ф'х

0 Ф2 •• ■ • Ф'2Х2

-^2 • .. h - ТП=1ф[хг

:i.4)

s

s

a

2. Dissipativity

A compact set in a phase space is called a Levinson set, if it is positively invariant and has a base of globally absorbing neighbourhoods.

A typical problem is to construct a good Levinson set. Below we give one result in this direction. Let T(O') be the tangent plane to the unstable manifold Wu(0') in the point 0'. We introduce also a couple of technical conditions:

Di : If s < 1, then h(s)^1(1) > -h'(s)(1 - s)fa(s). D2 : If s < 1, then ^i(s)^i(1) - ^l(1)^l(s) < 0.

Let us note that the conditions D1, D2 are satisfied for standard system (1.3).

Now let V0 is the simplex determined by the vertices O, O', qi, q2,..., qn, where

Фг(1) - h'(1)

(2.1)

Theorem 1. Let us consider system (1.1) satisfying conditions Ai-A5 and Di-D2.

The trajectories intersect the tangent plane T(O') transversally, except at O'. Moreover at any point, except at O', the trajectories intersect this plane in the direction inside a simplex V0, which is a Levinson set.

Proof. The Jacobian matrix at the point O' has the form

( фг (1) 0

0 Ф2 (1)

0 0

(2.2)

V -Фг(1) -Ф2(1) ... h'(1) )

where all the numbers on the diagonal are positive except h' (1). The tangent plane is defined by the equation

— + - + ... + — + s = 1, qi q2 qn

(2.3)

where qi are defined by (2.1), so that the vector orthogonal to this plane and directed outside the simplex can be written as

(q-1, 1, о

qi q2 qn

(2.4)

At any point of this plane the scalar product (let us denote it by A) of this vector with the vector of the system (1.1) is given by

A

( ( — ,..., —, 1 ) , \Ф1(8)Х1,... ,фп(в)Хп, h (s) - У^ фi(s)Xi ) \

\Vqi qn J \ ti )l

E

ф (s), qi

Xi

+ h(s) - ik(s)xi.

i=l i=l Consider now the family of the parallel planes

^ + ^ + ... + ^ + s = V, qi q2 qn

(2.5)

where V > 1. It can be checked directly, that if s = 1 then A < 0 except at O', where A = 0. In other points we get

A = h(s) + J]

i=i

Ms)

- Фi(s)

Xi

q

h(s) (1 - V) + -h(s-(V - s) + ^ X ^i(s) - q^(s)]

1 - s 1 - s

h(s) '

1s

(i - v ) + Yl

Xi

qi

h(s)

+ Фi(s) - qi^i(s)

<YX

q

- we use D1 here -

£

i=l

i=l

Xi

qi

i=1 h' (1)^i(s)

Ui)

1-s - qiA(s) + Фi(s)

<

Фi(s) ~Ui)

E

i=1

Xi

qi

(-Фг(1) + h' (1) - h' (1)) + Фi(s)

Фг(1)

By assumption D2 the proof is complete.

□

An example. Now, as in [2] we consider the system

with

Hence,

X = (тФ^) - d)x, s = h(s) - ф(s)x, s

h(s) = s(1 - s), ф(s)

(2.6)

s + a

m =1, a = 0.5, d = 0.25.

h' (1) = -1, Ф(1) = 7/12, ф(1) = 2/3 ^ q = 19/8.

Therefore the Levinson simplex in this case is defined by the straight line s = 1 — 8x/19 . The estimate in [2] gives the region bounded by lines s = 1, x + s = 5.

3. Degeneracy

Definition 1. An invariant set of system 1.1 will be called an inner set if its closure does not intersect with the boundary of R++1. If a system does not have inner sets it will be called degenerated.

Definition 2. The vector function F = F(s) = (F1,...,Fn) is called linearly determined on the interval I = [a,b], or the function collection F1,...,Fn is called linearly determined on I, if there exists a vector k = (k1,..., kn) such that (k, F (s)) < 0 for all s £ I and the inequality does not degenerate to equality on any subinterval I1 G I. A collection F1,... ,Fn which is not linearly determined will be called linearly connected.

Further we will consider only functions defined on the interval I = [0,1].

q

Theorem 2. If the function $ = ($i,..., $n) is linearly determined on I, then system, (1.1) is degenerated .

Proof. We suppose the opposite, that is, that there exists a trajectory (£,() = (£i;..., £n, Z), with closure wholly included in V = int^++i.

We consider the function L(x, s) = \n(xhk1 • • • x^') on V, where k = (ki,..., kn) is the vector making $ determined. The derivative of L along the trajectory (£,() equals L = (k,$(Z(t))) < 0 and, consequently, L is a Lyapunov function. The limit set of the trajectory (£,() is contained in the set W = {(x,s)\L = 0}. (This can be proved by standard methods. See for instance [10].)

But any connected component of W is contained in one hyperplane s = const > 0, which cannot contain whole trajectories except singular points. Because V does not contain any equilibirum the theorem is proved. □

We will now look at conditions for linear connectedness.

Statement 1. In the case n = 1, the function $ is linearly connected if and only if system (1.1) is degenerated.

Statement 2. Suppose n > 1. If there exists a linearly determined subcollection $j1,..., $jk, k < n for the collection $ = ($i,..., $n), then the collection $ is linearly determined.

Thus we can assumed that the functions are numbered according to condition A5. Remark also that $i(0) < 0, $i(1) > 0.

In connection to $ we consider the functions

t ft n t \ t( \ $i(s) < \ $i(s) ■ 1

f = (fi,...,fn), g = (9i,...,9n), Ji(s) = —77^, 9i(s) = —J^, 1 =1,...,n.

$i(0) $i(1)

Definition 3. We will say that scalar the functions p = p(s) and q = q(s) intersect at some point of the interval I, if in any neighbourhood of this point there exist si and s2, such that p(si) — q(si) < 0 and p(s2) — q(s2) > 0.

We will say that the functions p and q intersect on the interval I, if either there exists a point, where they intersect, or a subinterval Ii C I, where they are linearly dependent.

Statement 3. If the collection $ = ($i,... ,$n) is linearly connected, then any functions of the collections $, f and g intersect pairwise.

Proof. We assume the opposite. For example, that the functions fi and f2 do not intersect. This means that fi(s) — f2(s) = 0 on I. If fi(s) — f2(s) < 0 then

ki$i(s) + k2$2(s) > 0, ys E I, where ki = — , k2

$i(0) $2(0)

and the inequality does not degenerate to equality on any interval.

Choosing ki = 0 for i > 2, when n > 2, we get (k, $(s)) > 0, which is a contradiction. Statement is proved. □

In order to get explicit conditions for the linear connectedness of 0, we introduce more assumptions.

A6: The equation ki0i = kj0j does not have more than two solutions on I for any indices i,j, i = j, and any constants ki and kj, such that at least one of them is non-zero.

A7: The equation ci0i + Cj0j + ck0k = 0 does not have more than three solutions on the interval I for any indices i,j,k and for any constants ci, Cj, ck, such that at least one of them is non-zero.

s — X-

It is easily verified that the standard functions 0i =-- satisfy these conditions.

s + ai

We also notice that if all numbers ai are different, then the collection of functions of type (1.3) are not linearly dependent on any subinterval. We introduce the notations ai = gi(0), /i = fi(1) = a-i.

Theorem 3. We assume that the functions 0 satisfy the conditions A1 — A6. If 0 is linearly connected on I, then

1) 3i < / < ... < fin, ai > a2 > ... > an.

2) The functions fi, i = 1,... ,n, intersect pairwise exactly one time on (0,1), and if 9ij is the intersection point of fi and fj(i < j) , then 9ij E (Xi, 1).

Analogously, the functions gi, i = 1,... ,n, intersect pairwise exactly one time on (0,1), and if Tij is the intersection point of the functions gi and gj(i < j) , then

Tij E (0, Xj).

3) If n > 2, and the function 0 satisfy condition A7 and 9ik and 9jk are the intersection points of fi and fj with fk and i < j < k, then 9ik > 9jk.

Analogously, if Tik and Tjk are intersection points of gi, gj with gk, and i < j < k, then Tik > Tjk.

Proof. Statement 2) follows from the pairwise linearly connectedness of the functions 0i and 0j. We show this for the functions f1 and f2. Clearly these functions must have an intersection and according to A6 only one. We suppose the intersection is not in Ii = (Xi, 1). Then f2 <fi in Ii.

-1 - 2

У • 9i(s)

1 ■ V*^ 1 к

-1 ■ j'Y \ * Ai 1 s

-2 ■

-3 ■

- 4 ■

Рис. 1. Ai = 0.6, A2 = 0.35, a1 = 0.55, a2 = 0.1 We consider the minimal k > 0 such, that kf2(s) < fi(s) for all s E I\. Evidently ISSN 0203-3755 Динамические системы, 2016, том 6(34), №1

k < 1. From A6 and kf2(0) < f1(0) and kf2(X2) < f1(X2) follows that the same inequality is satisfied in s E [0, X2].

The inequality is automatically satisfied in the segment [X^X^. We obtain that kf2(s) < f1(s) everywhere in I, which contradicts the linear connectedness of fi. Analogously, 2) is proved for any pair from collection g. Further:

Xl > X2 ^ fl(Xi) >f2(Xi) ^ fl(1) <f2 (1) ^ Pi <02.

We get similar inequalities for the functions g1, g2 from which follows a1 > a2. Thus 1) is proved.

We now prove 3) for any three functions from collection f. For example, for f1, f2, f3. We assume the opposite, that is, 913 < 623. We define c° = f2(X3)/f1(X3). Clearly, 0 < c° < 1. We consider the function lc(s) = f2(s) — cf1(s) — (1 — c)f3(s). It is clear, that lc(0) = 0 and lc(013) < 0 for any c E [0,1]. Moreover, lco(X3) = 0.

We consider the intervals on which lco(s) > 0 (if such do not exist, proof is finished). According to A7 the function lco (s) does not have more than three zeros and thus there are no more than three of such intervals, and if they are two, then their closure contains X3.

In this case there exists a c < c0 such that again there are two of these intervals, but lc(X3) < 0, and thus their closure does not contain X3.

Рис. 2. Ai = 0.6, A2 = 0.35, A3 = 0.25, a1 = 0.55, a2 = 0.1, a3 = 0.05

Consequently lc does not have less than four zeros, which contradicts assumption. It remains to consider the case, when there are only one interval of positiveness. We note that this should be satisfied for all c E [0, c°\. Moreover the interval of positiveness intersect for nearby values of parameters. Consequently the interval should be on the left for c =0, because the interval is to left of 0\3 for c = c0.

On the other side, the inequality l0(1) > 0 is satisfied and we get a contradiction proving 3). Theorem is proved. □

We now consider sufficient conditions.

Theorem 4. Suppose function 0 satisfies conditions A1 — A5. Then

in the case when n = 2, one of the conditions [1 > [2 or a1 > a2 is sufficient for linear connectedness;

for the case n = 3, one of conditions [1 > [2 > [3, > @23 or a1 > a2 >

a3, t13 < t23 is sufficent for linear connectedness.

Proof. Let n = 2 and [1 > [2. We suppose that the function f is linearly determined, that is that k1f1(s) + k2f2(s) < 0 for some vector k, \k\ = 0. Substituting s = 0, A2, A^ 1, we get a contradiction with the inequality for k, which proves this part of the theorem.

Let n = 3 and [1 > [2 > [3. We assume that the function f is linearly determined, that is that k1f1(s) + k2f2(s) + k3f3(s) < 0 for some vector k, \k\ = 0. Substituting s = 0, we get k1 + k2 + k3 > 0. Substituting s = 623, we get k1j + k2 + k3 < 0, where fi = f1(623)/f2(623) < 1, from which follows k1 > 0.

Analogously, substituting 012, we get k3 > 0 and, automatically, k2 < 0.

On the other side, for s = 913 we get (k1 + k3)f1 (913) + k2f2(613) < 0. But then 0 < f1(013) < f2(913), leads to a contradiction proving the theorem. □

4. Standard three dimensional system (n=2)

We use here some ideas and methods of [12], where sufficient conditions for degeneracy of the standard system were obtained.

Explicit formulas for degeneracy in the standard system were obtained in [?]-[12] for n = 2 and moreover theorem 2 was proved for this system. Concretely it was shown that, when A2 < A1, if one of the following conditions

1) a1 < a2,

2) a1A2 — a2A1 < 0,

3) aia2(Ai — A2) + A1A2(ai — a2) > aiA2 — a2Ai,

is satisfied then one of the coordinate plane is globally attracting in

We introduce the notations

A2ai 1 — A2 1 + ai

Y =-, a =-•-.

Aa 1 + a2 1 — Ai

In case A1 > A2 it follows directly from theorems 3-4 that the function

0(s) = (0i(sU(s))=(^, ^)

s + a1 s + a2

is linearly connected if 7 > a.

We show that then a > 1. We first prove that a1 > a2. Indeed, let a1 < a2. Then A2 a1 A1 a2 A2a1 1 — A2 1 + a1

< --r- • -- T- <

1 — A2 1 + a1 1 — A1 1 + a2 A1a2 1 — A1 1 + a2 We got a contradiction.

Further:

\ \ 1 " 1 c 1 + ai

Ai > X2, ai > a2 ^-— > 1 & - > 1 ^ a > 1.

1 — Ai 1 + a2

Thus the condition A1 > A2 is equivalent to y > a > 1 in the case of linear connectedness.

Simple transforms show that the complement to these inequalities conicides with the set of parameters given by 1) — 3).

5. Conclusion

We have obtained conditions for dissipativity and degeneracy (meaning extinction of some predators) for a general family of predator-prey systems with many predators and one prey. Considering degeneracy we have only used the properties of the equations of the predators. Using the equation for the prey will give better results. Finally we include figure 3 for the standard three dimensional system showing results of numerical experiments for the regions of degeneracy of one predators and for existence of inner solutions. In region 1 predator x2 goes extinct and x\ in region 4. An inner solution exists in regions 2 and 3 and on the boundary between them there is a period doubling bifurcation. The regions are shown in the parameter plane of parameters Ai and X2 keeping a1 and a2 fixed.

Рис. 3. Regions of extinction and coexistence for a1 = 0.2, a2 = 0.02

We observe that condition 1) in previous section implies condition 2) which implies condition 3) and condition 3) implies x1 ^ 0. Condition 3) in the figure gives a subregion of region 4. We also observe that changing the order of variables we get

analogous conditions in the case when X1 < X2 and then corresponding condition 1) gives x2 ^ 0 for a1 > a2. This corresponds to a subregion of region 1.

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Получена 03.03.2016