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Survival and extinction in competitive systems
Nonlinear Analysis: Real World Applications 9 (2008) 708 – 717 www.elsevier.com/locate/na
Survival and extinction in competitive systems Shair Ahmada,∗ , Ivanka Stamovab a Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA b Bourgas Free University, 8000 Bourgas, Bulgaria
Received 11 December 2006; accepted 11 December 2006
Abstract In [S. Ahmad, I.M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. Real World Appl. 5 (2004)], implications of various inequalities involving interaction coefficients and averages of the growth rates were listed for a two-dimensional Lotka–Volterra system, and it was shown that the first condition can be extended to one that is almost necessary and sufficient for persistence of a three-dimensional Lotka–Volterra system. In this paper, we give similar extensions of all the cases. We also explore the relationship of the (I, J ) condition, a notion defined by Ahmad–Lazer recently (see [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006)]), to certain known conditions which imply persistence. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Extinction; Persistence; Permanence; Growth rate; Interaction coefficients; (I, J ) condition
1. Introduction and summary The property of strong persistence for ecological models based on systems of differential equations of the form uk (t) = uk (t)gk (u1 (t), . . . , uN (t)), k = 1, . . . , N,
(1.1)
has been widely investigated. For extensive bibliographies and historical remarks, see [10,14,15]. Ref. [14] contains discussions of characterizations of strong persistence in terms of the topological structure of the orbits of the dynamical system defined by (1.1) on the set (R+ )N = {col(x1 , . . . , xN )|xk 0, 1 k N }, as well as in terms of the existence of certain types of Lyapunov functions. (It is well known that (R+ )N is positively invariant with respect to (1.1).) For related and supplemental reading one is referred to [7–9,11–13]. Following Refs. [10] and [14], we have Definition 1.1. System (1.1) is said to be strongly persistent if for any solution u=col (u1 , . . . , uN ) with uk (t0 ) > 0, k= 1, . . . , N, u is defined on [t0 , ∞) and for k = 1, . . . , N lim inf uk (t) > 0. t→∞
System (1.1) is called persistent if “lim inf” is replaced by “lim sup” in this definition. ∗ Corresponding author.
E-mail address: [email protected] (S. Ahmad). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.12.011
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Definition 1.2. System (1.1) is called permanent if there exist constants d and R such that if u is any solution such that uk (t0 ) > 0, k = 1, 2, . . . , N, then there exists t∗ = t∗ (u) such that t t∗ (u) implies that d uk (t) R, k = 1, 2, . . . , N. The system is called strongly permanent if small perturbations of it are permanent. In [4], we considered a narrower type of strong persistence for a class of competitive Lotka–Volterra systems which may be nonautonomous. For k =1, . . . , N, let ak (t) be continuous for t ∈ (−∞, ∞)=R with 0 < akL ≡ inf R ak (t) supR ak (t) ≡ akM < ∞. We also assume that for k = 1, . . . , N, the limit 1 t0 +T ak (t) dt ≡ M[ak ] lim T →∞ T t0 exists, uniformly with respect to t0 ∈ (−∞, ∞). This will be true, for example, if the functions are almost periodic or periodic. We consider the system N uk (t) = uk (t) ak (t) − bk u (t) , (LV) =1
k = 1, . . . , N, where for 1 k, N, bk is a constant with bk 0 and bkk > 0 for k = 1, . . . , N. Definition 1.3. We shall call system (LV) totally persistent (totally permanent) if for any nonempty subset I of {1, . . . , N}, the subsystem ⎫ ⎪ ⎪ ⎬ ui (t) = ui (t) ai (t) − bij uj (t) (LV-I ) ⎪ j ∈I ⎪ ⎭ i∈I is strongly persistent (strongly permanent). If (LV) is interpreted as a model for competition between N species, then total persistence means that not only is this system strongly persistent, but also any smaller system which results by removing some of the species by setting their initial values equal to zero is also strongly persistent. (The uniqueness theorem implies that if uj (t0 ) = 0 for some j ∈ {1, . . . , N}, then uj (t) ≡ 0.) The simplest type of (LV)-system is the autonomous logistic equation: u (t) = u(t)[a − bu(t)], where a and b are positive constants. If u(0) > 0, then u(t) → a/b as t → ∞, so this system is totally persistent. For N = 2, total persistence of an autonomous, competitive (LV)-system is equivalent to strong persistence, since each of the systems obtained by setting one component equal to zero is a logistic equation. However for N = 3, strong persistence of an autonomous, competitive (LV)-system does not imply total persistence, as the following example shows (see [6]). Consider the three-dimensional Lotka–Volterra system u = u[150 − 50u − 100v − w], v = v[400 − u − 200v − 200w], w = w[105 − u − v − 100w], where u(0) > 0, v(0) > 0 and w(0) > 0. It can be verified that this system is strongly persistent. However, it is not totally persistent since the smaller system u (t) = u(t)[150 − 50u(t) − 100v(t)], v (t) = v(t)[400 − u(t) − 200v(t)],
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which is obtained from the larger system by setting w(t) ≡ 0 is not persistent. In fact, if u(0) > 0, v(0) > 0, then it follows that u(t) → 0 as t → ∞ exponentially. Definition 1.4. System (LV) satisfies the (I, J )-conditions if whenever I is a proper subset of {1, . . . , N} for which there exists a solution of the linear system M[ai ] = bi x , i ∈ I , (1.2) ∈I
with xi > 0 ∀i ∈ I , then for J = {1, . . . , N} − I we have ∀j ∈ J,
M[aj ] >
bj x .
(1.3)
∈I
In [4], it was shown that (I, J ) conditions are both necessary and sufficient for total permanence. 2. Relationship of (I, J ) condition to certain known conditions implying persistence In [2] the authors considered systems in which the interaction coefficients as well as the growth rates could vary. If we specialize the result of [2] to the case where the interaction coefficients are constant and where the growth rates satisfy the assumptions given in the introductory section, then it says: If for i = 1, . . . , N M[ai ] >
N
bij M[aj ]/bjj
(2.1)
j =1
j =i
then (LV) is strongly persistent. Inequalities (2.1) have played an important role in studying persistence, permanence, and extinction (see [1–6]), and we will consider them in the next section. We now discuss the relationship of (2.1) to the (I, J ) condition. It is not difficult to see that inequalities (2.1) imply that the (I, J )-conditions hold for (LV). To see this, let I be a proper subset of {1, . . . , N} and let J be the complimentary set. Assume that for i ∈ I, xi > 0 and for all i ∈ I M[ai ] = bik xk . k∈I
Since for all i, k ∈ I, bik 0 and for all i ∈ I, bii > 0, it follows that ∀i ∈ I, xi < M[ai ]/bii . Let j ∈ J . From (2.1) and the above, we see that since I ∩ J = , M[aj ] >
N =1
=j
bj M[a ]/b
∈I
bj M[a ]/b
bj x
∈I
which proves the assertion. For an example of a system of the type which satisfies the (I, J )-conditions but not conditions (2.1), we consider the highly symmetrical system u1 = u1 [4 − 4u1 − 3u2 − 3u3 ], u2 = u2 [4 − 3u1 − 4u2 − 3u3 ], u3 = u3 [4 − 3u1 − 3u2 − 4u4 ].
(2.2)
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Since for i = 1–3 M[ai ] = ai = 4 < 6 =
3
bij M[aj ]/bjj
j =1
j =i
none of conditions (2.1) hold. Let I = {1, 2}. The system a1 = 4 = b11 x1 + b12 x2 = 4x1 + 3x2 , a2 = 4 = b21 x1 + b22 x2 = 3x1 + 4x2 has the unique solution x1 = Since
4 7
= x2 .
a3 = 4 > 3(4/7) + 3(4/7) = b31 x1 + b32 x2 the (I, J )-condition corresponding to I = {1, 2} is satisfied. Therefore, because of the symmetry of (2.2) the (I, J )conditions corresponding to I = {1, 3} or I = {2, 3} are also satisfied. Let I = {1}. The equation: a1 = 4 = b11 x1 = 4x1 has the unique solution x1 =1. Since for j =2 or 3, the condition aj > bj 1 x1 is 4 > 3, the (I, J )-conditions corresponding to I = {1} are satisfied. Again by symmetry of (2.2), the (I, J )-conditions are satisfied for I = {2} or I = {3}. This shows that (2.2) satisfies the (I, J )-conditions. In the above example the constant growth rates can be replaced by variable growth rates with averages equal to 4. Also since the (I, J )-conditions are stable under small perturbations of the interaction coefficients and growth rate averages, it is easy to see that there are nonsymmetrical (LV)-systems which satisfy the (I, J )-conditions but none of conditions (2.1). 3. Almost necessary and sufficient conditions for persistence when n = 3 We consider system (LV) when n = 3 and give conditions under which if u(t) = (u1 (t), u2 (t), u3 (t)) is a solution with uk (t0 ) > 0, k = 1–3, then lim sup uk (t) > 0
(3.1)
t→∞
for k = 1–3. These conditions will involve certain inequalities among quantities depending on the averages of the growth rates and the interaction coefficients. These inequalities, which are strict, are sufficient for (3.1) to hold and are also almost necessary in the sense that if any one of them is reversed, then (3.1) does not hold. Throughout this section we assume that if i = j then M[ai ] = bij M[aj ]/bjj .
(3.2)
As our main result will show, systems in which this is not true are structurally unstable in the sense that slight changes in the averages of the growth rates or the interaction coefficients can drastically alter the long-term behavior of solutions. The following theorem concerning the two-species system u1 (t) = u1 (t)[a1 (t) − b11 u1 (t) − b12 u2 (t)], u2 (t) = u2 (t)[a2 (t) − b21 u1 (t) − b22 u2 (t)],
(LV2 )
where the growth rates and interaction coefficients satisfy the assumptions of the previous sections, appeared in [5].
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Theorem 3.1. Let (u1 (t), u2 (t)) be a solution of (LV2 ) such that u1 (t0 ) > 0, u2 (t0 ) > 0. For i = 1, 2, and T > 0 let 1 t0 +T Ai (T ) = ui (t) dt. T t0 (a) If M[a1 ] > b12 M[a2 ]/b22 , M[a2 ] > b21 M[a1 ]/b11 , then for i = 1, 2, lim Ai (T ) = xi∗ ,
T →∞
where x1 = x1∗ , x2 = x2∗ is the unique solution of b11 x1 + b12 x2 = M[a1 ], b21 x1 + b22 x2 = M[a2 ].
(3.3)
(b) If M[a1 ] > b12 M[a2 ]/b22 , M[a2 ] < b21 M[a1 ]/b11 , then lim A1 (T ) = M[a1 ]/b11 ,
T →∞
lim A2 (T ) = 0.
T →∞
(c) If M[a1 ] < b12 M[a2 ]/b22 , M[a2 ] > b21 M[a1 ]/b11 , then lim A1 (T ) = 0,
T →∞
lim A2 (T ) = M[a2 ]/b22 .
T →∞
(d) If M[a1 ] < b12 M[a2 ]/b22 , M[a2 ] < b21 M[a1 ]/b11 , then either (i) A1 (T ) → M[a1 ]/b11 , A2 (T ) → 0 as T → ∞, (ii) A1 (T ) → 0, A2 (T ) → M[a2 ]/b22 as T → ∞, or ∗ (iii) there exists a sequence {Tn }∞ 1 such that Tn+1 > Tn > 0 for n 1, Tn → ∞ as n → ∞ and Ai (Tn ) → xi as ∗ ∗ n → ∞, i = 1, 2, where x1 = x1 , x2 = x2 is the unique solution of (3.3). For proofs of the following two theorems see [5,6]. Theorem 3.2. Consider an (LV) system in which the number of species N = m + r, where m 1, r 1, and assume that the (I, J ) conditions are satisfied in the m-species (LV) system ui (t) = ui (t)[ai (t) −
m
bi u (t)],
1 i m.
=1
Let xl > 0, l = 1, . . . , m, be the unique positive numbers (see [4]) such that for i = 1, . . . , m, M[ai ] =
m
bil xl
l=1
and assume that for j = m + 1, . . . , N, M[aj ] <
m
bj l xl .
l=1
Let A be a compact subset of {(1 , . . . , m )|i > 0, i = 1, . . . , m}. There exists a number d =d(A) > 0 such that if u=col(u1 , . . . , uN ) is any solution of (LV), with (u1 (t0 ), . . . , um (t0 )) ∈ A and 0 < uj (t0 )d(A), j = m + 1, . . . , N, then for j = m + 1, . . . , N, lim uj (t) = 0.
t→∞
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Theorem 3.3. System (LV)3 is persistent if for any permutation of i, j, k of 1,2,3, M[ai ] > bij M[aj ]/bjj , M[aj ] > bj i M[ai ]/bii , M[ak ] > bki xi∗ + bkj xj∗ , where xi = xi∗ , xj = xj∗ is the unique solution of M[ai ] = bii xi + bij xj , M[aj ] = bj i xi + bjj xj . Furthermore, if the first two inequalities hold and the third one is reversed, i.e. M[ak ] < bki xi∗ + bkj xj∗ then the system is not persistent. The above theorem is incomplete in the sense that it corresponds to part (a) of Theorem 3.1, and does not explain what the implications of the remaining parts of this theorem might be. Our next theorem gives a complete classification based on Theorem 3.1. The following result will facilitate the proof of our main theorem. Lemma 3.1. If conditions (d) of Theorem 3.1 hold, M[a3 ] > b31 M[a1 ]/b11 , and M[a3 ] > b32 M[a2 ]/b22 , then M[a3 ] > b31 x1∗ + b32 x2∗ . Proof. We have that 1 = b11
x1∗ x∗ + b12 2 . M[a1 ] M[a1 ]
Since, by (d), b12 b22 > M[a1 ] M[a2 ] we see that 1>
b11 x1∗ b22 x2∗ + . M[a1 ] M[a2 ]
Consequently, by the hypotheses of the lemma b31 b32 ∗ ∗ 1 > x1 + x2 M[a3 ] M[a3 ] and the lemma is proved.
Theorem 3.4. Assume that for ⎡ ⎤ 3 ui (t) = ui (t) ⎣ai (t) − bij uj (t)⎦ ,
i = 1, 2, 3,
(LV3 )
j =1
conditions (3.2) hold for 1i, j 3 and i = j . This system will be persistent if the following conditions are satisfied. Let i, j, k be any permutation of 1, 2, 3. (a) If M[ai ] > bij M[aj ]/bjj and M[aj ] > bj i M[ai ]/bii , then M[ak ] > bki xi∗ + bkj xj∗ ,
(3.4)
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where xi = xi∗ , xj = xj∗ is the unique solution of M[ai ] = bii xi + bij xj , M[aj ] = bj i xi + bjj xj .
(3.5)
(b) If M[ai ] > bij M[aj ]/bjj , and M[aj ] < bj i M[ai ]/bii , then M[ak ] > bki M[ai ]/bii .
(3.6)
(c) If M[ai ] < bij M[aj ]/bjj and M[aj ] > bj i M[ai ]/bii , then M[ak ] > bkj M[aj ]/bjj .
(3.7)
(d) If M[ai ] < bij M[aj ]/bjj and M[aj ] < bj i M[ai ]/bii , then both of the inequalities (3.6) and (3.7) hold. These conditions are sharp in the following sense: If the conditions of (a) hold and M[ak ] < bki xi∗ + bkj xj∗
(3.8)
then (LV3 ) is not persistent. If the conditions of (b) hold and M[ak ] < bki M[ai ]/bii
(3.9)
then (LV3 ) is not persistent. If the conditions of (c) hold and M[ak ] < bkj M[aj ]/bjj
(3.10)
then (LV3 ) is not persistent. If the conditions of (d) hold and either one of the inequalities (3.9) or (3.10) is satisfied, then (LV3 ) is not persistent. If i, j, k is a fixed permutation of 1,2,3 then (a)–(c) are sharp sufficient conditions for persistence of the kth species, i.e., lim sup uk (t) > 0. t→∞
Proof. If (LV3 ) is not persistent then we have a solution (u#1 (t), u#2 (t), u#3 (t)) of (LV3 ) such that u#j (t0 ) > 0 for j = 1–3 and for some k, 1k 3, u#k (t) → 0 as t → ∞. By permutation of indices we may assume that k = 3. By replacing t0 by a larger number, if necessary, we may assume that a (t) − b3 u#3 (t) > 0 for t t0 − 1 and = 1, 2. If we define for = 1, 2 t t0 , a (t) − b3 u#3 (t), # a (t) = a (t) − (t − t0 + 1)b3 u#3 (t), t0 − 1 t < t0 , t < t0 − 1, a (t), # > 0 and a # < ∞ for = 1, 2. Moreover, since u# (t) → 0 as t → ∞, the then a# is continuous on (−∞, ∞), aL 3 M limit 1 t∗ +T # 1 t∗ +T M[a# ] = lim a (t) dt = lim a (t) dt = M[a ] T →∞ T t∗ T →∞ T t∗
exists uniformly with respect to t∗ ∈ (−∞, ∞) and = 1, 2. For proof of part (a), see [5,6]. Suppose the conditions of part (b) hold. We may assume i = 1. Then M[a1# ] = M[a1 ] > b12
M[a2# ] M[a2 ] = b12 b22 b22
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and M[a2# ] = M[a2 ] < b21
M[a1# ] M[a1 ] = b21 . b11 b11
Therefore, by part (b) of Theorem 3.1 applied to (LV#2 ), we see that 1 T →∞ T
t0 +T
lim
t0
u#1 (t) dt =
M[a1 ] M[a1# ] = b11 b11
and 1 lim T →∞ T
t0 +T
t0
u#2 (t) dt = 0.
Since (3.6) holds for k = 3, i = 1, and since u#3 (t) → 0 as t → ∞,
1 T →∞ T
t0 +T
lim
(a3 (t) −
t0
3
b3 u# (t)) dt
=1
= M[a3 ] − b31 M[a1 ]/b11 > 0. But since u#3 (t) → 0 as t → ∞, it follows that for T large 1 T
t0 +T
(a3 (t) −
t0
3
b3 u# (t)) dt
=1
u#3 (t0 + T ) 1 = ln <0 T u#3 (t0 )
so we again have a contradiction. This proves part (b). The proof for part (c) is the same as that for part (b) with the roles of u#1 (t) and u#2 (t) interchanged. Suppose that the conditions of part (d) are satisfied. Since M[a1# ] = M[a1 ] < b12 M[a2 ]/b22 = b12 M[a2# ]/b22 and M[a2# ] = M[a2 ] < b21 M[a1 ]/b11 = b21 M[a1# ]/b11 and u1 (t) = u#1 (t), u2 (t) = u#2 (t) is a solution of (LV#2 ) for t t0 , it follows from part (d) of Theorem 3.1 that either t +T lim 1 0 u#1 (t) dt = M[a1 ]/b11 , T →∞ T t0 t +T (3.11) lim T1 t00 u#2 (t) dt = 0, T →∞
or
1 T →∞ T lim 1 T →∞ T
lim
t0 +T t0 t0 +T t0
u#1 (t) dt = 0, u#2 (t) dt = M[a2 ]/b22 ,
(3.12)
or there exists a sequence {Tn }∞ 1 such that Tn+1 > Tn > 0 for n 1, Tn → ∞ as n → ∞ and for = 1, 2 1 n→∞ Tn
t0 +Tn
lim
t0
u# (t) dt = x∗ .
(3.13)
If (3.11) holds, then since, by assumption, M[a3 ] > b31 M[a1 ]/b11 , we get a contradiction just as we did in the proof of part (b). The case in which (3.12) holds also leads to a contradiction as in the proof of part (c).
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If (3.13) holds, then since u#3 (t) → 0 as t → ∞, it follows from the lemma that t0 +Tn 3 1 # lim a3 (t) − b3 u (t) dt = M[a3 ] − b31 x1∗ − b32 x2∗ > 0. n→∞ Tn t 0 =1
However, for n large t0 +Tn 3 u#3 (t0 + Tn ) 1 1 # a3 (t) − b3 u (t) dt = n < 0, Tn Tn t0 u#3 (t0 ) =1
so we have a contradiction. This proves that the conditions of the theorem are sufficient for persistence of (LV3 ). Case (a), where M[a3 ] < b31 x1∗ + b32 x2∗ , assuming that k = 3, was handled in [5,6]. Suppose that in case (b) i = 1, j = 2, k = 3 and M[a3 ] < b31 M[a1 ]/b11 . Obviously the single species (LV)-system u1 (t) = u1 (t)[a1 (t) − b11 u1 (t)] satisfies the (I, J )-conditions. Since the unique solution x1 = x1∗ of b11 x1 = M[a1 ] is x1∗ = M[a1 ]/b11 and since, for j = 2, 3, M[aj ] < bj 1 M[a1 ]/b11 = bj 1 x1∗ , it follows from Theorem 3.2 (with m = 1) that there are solutions of (LV3 ), u = (u1 , u2 , u3 ), u1 (t0 ) > 0, u2 (t0 ) > 0, u3 (t0 ) > 0 with u2 (t) → 0, u3 (t) → 0 as t → ∞. Therefore (LV3 ) is not persistent. Similarly, if M[a1 ] < b12 M[a2 ]/b22 , M[a3 ] < b32 M[a2 ]/b22 , then there are solutions of (LV3 ) with u (t0 ) > 0, = 1–3 and u1 (t) → 0, u3 (t) → 0 as t → ∞. This together with the above shows that when conditions (b) hold, (3.9) precludes persistence, and when conditions (c) hold, (3.10) precludes persistence. The same argument shows that when conditions (d) hold either (3.9) or (3.10) precludes persistence. The last statement of the theorem is obvious. References [1] S. Ahmad, A.C. Lazer, Necessary and sufficient average growth in a Lotka–Volterra system, Nonlinear Anal. 34 (1998). [2] S. Ahmad, A.C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka–Volterra system, Nonlinear Anal. 40 (2000). [3] S. Ahmad, A.C. Lazer, Average growth and extinction in a competitive Lotka–Volterra system, Nonlinear Anal. 62 (2005). [4] S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006). [5] S. Ahmad, I.M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. Real World Appl. 5 (2004). [6] S. Ahmad, I.M. Stamova, Partial persistence and extinction in N -dimensional competitive systems, Nonlinear Anal. 60 (2005). [7] R. Bellman, Matrix Analysis, second ed., McGraw-Hill, New York, 1975. [8] T. Hallam, L. Svoboda, T. Gard, Persistence and extinction in three species Lotka–Volterra competitive systems, Math. Biosci. 46 (1979). [9] P. Hartman, Ordinary Differential Equations, P. Hartman, Baltimore, 1973. [10] J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, New York, 1988.
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[11] W. Jansen, A permanence theorem for replicator and Lotka–Volterra systems, J. Math. Biol. 25 (1986). [12] R.M. May, W. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975). [13] R. Redheffer, Lotka–Volterra systems with constant interaction coefficients, Nonlinear Anal. 46 (2001). [14] Y. Takeuchi, Global Dynamical Properties of Lotka–Volterra Systems, World Scientific, Singapore, 1996. [15] P. Waltman, A brief survey of persistence in dynamical systems, in: Busenberg, M. Martelli (Eds.), Delay Differential Equations and Dynamical Systems, fifth ed., Springer, Berlin, 1991.
Survival and extinction in competitive systems Shair Ahmada,∗ , Ivanka Stamovab a Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA b Bourgas Free University, 8000 Bourgas, Bulgaria
Received 11 December 2006; accepted 11 December 2006
Abstract In [S. Ahmad, I.M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. Real World Appl. 5 (2004)], implications of various inequalities involving interaction coefficients and averages of the growth rates were listed for a two-dimensional Lotka–Volterra system, and it was shown that the first condition can be extended to one that is almost necessary and sufficient for persistence of a three-dimensional Lotka–Volterra system. In this paper, we give similar extensions of all the cases. We also explore the relationship of the (I, J ) condition, a notion defined by Ahmad–Lazer recently (see [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006)]), to certain known conditions which imply persistence. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Extinction; Persistence; Permanence; Growth rate; Interaction coefficients; (I, J ) condition
1. Introduction and summary The property of strong persistence for ecological models based on systems of differential equations of the form uk (t) = uk (t)gk (u1 (t), . . . , uN (t)), k = 1, . . . , N,
(1.1)
has been widely investigated. For extensive bibliographies and historical remarks, see [10,14,15]. Ref. [14] contains discussions of characterizations of strong persistence in terms of the topological structure of the orbits of the dynamical system defined by (1.1) on the set (R+ )N = {col(x1 , . . . , xN )|xk 0, 1 k N }, as well as in terms of the existence of certain types of Lyapunov functions. (It is well known that (R+ )N is positively invariant with respect to (1.1).) For related and supplemental reading one is referred to [7–9,11–13]. Following Refs. [10] and [14], we have Definition 1.1. System (1.1) is said to be strongly persistent if for any solution u=col (u1 , . . . , uN ) with uk (t0 ) > 0, k= 1, . . . , N, u is defined on [t0 , ∞) and for k = 1, . . . , N lim inf uk (t) > 0. t→∞
System (1.1) is called persistent if “lim inf” is replaced by “lim sup” in this definition. ∗ Corresponding author.
E-mail address: [email protected] (S. Ahmad). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.12.011
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Definition 1.2. System (1.1) is called permanent if there exist constants d and R such that if u is any solution such that uk (t0 ) > 0, k = 1, 2, . . . , N, then there exists t∗ = t∗ (u) such that t t∗ (u) implies that d uk (t) R, k = 1, 2, . . . , N. The system is called strongly permanent if small perturbations of it are permanent. In [4], we considered a narrower type of strong persistence for a class of competitive Lotka–Volterra systems which may be nonautonomous. For k =1, . . . , N, let ak (t) be continuous for t ∈ (−∞, ∞)=R with 0 < akL ≡ inf R ak (t) supR ak (t) ≡ akM < ∞. We also assume that for k = 1, . . . , N, the limit 1 t0 +T ak (t) dt ≡ M[ak ] lim T →∞ T t0 exists, uniformly with respect to t0 ∈ (−∞, ∞). This will be true, for example, if the functions are almost periodic or periodic. We consider the system N uk (t) = uk (t) ak (t) − bk u (t) , (LV) =1
k = 1, . . . , N, where for 1 k, N, bk is a constant with bk 0 and bkk > 0 for k = 1, . . . , N. Definition 1.3. We shall call system (LV) totally persistent (totally permanent) if for any nonempty subset I of {1, . . . , N}, the subsystem ⎫ ⎪ ⎪ ⎬ ui (t) = ui (t) ai (t) − bij uj (t) (LV-I ) ⎪ j ∈I ⎪ ⎭ i∈I is strongly persistent (strongly permanent). If (LV) is interpreted as a model for competition between N species, then total persistence means that not only is this system strongly persistent, but also any smaller system which results by removing some of the species by setting their initial values equal to zero is also strongly persistent. (The uniqueness theorem implies that if uj (t0 ) = 0 for some j ∈ {1, . . . , N}, then uj (t) ≡ 0.) The simplest type of (LV)-system is the autonomous logistic equation: u (t) = u(t)[a − bu(t)], where a and b are positive constants. If u(0) > 0, then u(t) → a/b as t → ∞, so this system is totally persistent. For N = 2, total persistence of an autonomous, competitive (LV)-system is equivalent to strong persistence, since each of the systems obtained by setting one component equal to zero is a logistic equation. However for N = 3, strong persistence of an autonomous, competitive (LV)-system does not imply total persistence, as the following example shows (see [6]). Consider the three-dimensional Lotka–Volterra system u = u[150 − 50u − 100v − w], v = v[400 − u − 200v − 200w], w = w[105 − u − v − 100w], where u(0) > 0, v(0) > 0 and w(0) > 0. It can be verified that this system is strongly persistent. However, it is not totally persistent since the smaller system u (t) = u(t)[150 − 50u(t) − 100v(t)], v (t) = v(t)[400 − u(t) − 200v(t)],
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which is obtained from the larger system by setting w(t) ≡ 0 is not persistent. In fact, if u(0) > 0, v(0) > 0, then it follows that u(t) → 0 as t → ∞ exponentially. Definition 1.4. System (LV) satisfies the (I, J )-conditions if whenever I is a proper subset of {1, . . . , N} for which there exists a solution of the linear system M[ai ] = bi x , i ∈ I , (1.2) ∈I
with xi > 0 ∀i ∈ I , then for J = {1, . . . , N} − I we have ∀j ∈ J,
M[aj ] >
bj x .
(1.3)
∈I
In [4], it was shown that (I, J ) conditions are both necessary and sufficient for total permanence. 2. Relationship of (I, J ) condition to certain known conditions implying persistence In [2] the authors considered systems in which the interaction coefficients as well as the growth rates could vary. If we specialize the result of [2] to the case where the interaction coefficients are constant and where the growth rates satisfy the assumptions given in the introductory section, then it says: If for i = 1, . . . , N M[ai ] >
N
bij M[aj ]/bjj
(2.1)
j =1
j =i
then (LV) is strongly persistent. Inequalities (2.1) have played an important role in studying persistence, permanence, and extinction (see [1–6]), and we will consider them in the next section. We now discuss the relationship of (2.1) to the (I, J ) condition. It is not difficult to see that inequalities (2.1) imply that the (I, J )-conditions hold for (LV). To see this, let I be a proper subset of {1, . . . , N} and let J be the complimentary set. Assume that for i ∈ I, xi > 0 and for all i ∈ I M[ai ] = bik xk . k∈I
Since for all i, k ∈ I, bik 0 and for all i ∈ I, bii > 0, it follows that ∀i ∈ I, xi < M[ai ]/bii . Let j ∈ J . From (2.1) and the above, we see that since I ∩ J = , M[aj ] >
N =1
=j
bj M[a ]/b
∈I
bj M[a ]/b
bj x
∈I
which proves the assertion. For an example of a system of the type which satisfies the (I, J )-conditions but not conditions (2.1), we consider the highly symmetrical system u1 = u1 [4 − 4u1 − 3u2 − 3u3 ], u2 = u2 [4 − 3u1 − 4u2 − 3u3 ], u3 = u3 [4 − 3u1 − 3u2 − 4u4 ].
(2.2)
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Since for i = 1–3 M[ai ] = ai = 4 < 6 =
3
bij M[aj ]/bjj
j =1
j =i
none of conditions (2.1) hold. Let I = {1, 2}. The system a1 = 4 = b11 x1 + b12 x2 = 4x1 + 3x2 , a2 = 4 = b21 x1 + b22 x2 = 3x1 + 4x2 has the unique solution x1 = Since
4 7
= x2 .
a3 = 4 > 3(4/7) + 3(4/7) = b31 x1 + b32 x2 the (I, J )-condition corresponding to I = {1, 2} is satisfied. Therefore, because of the symmetry of (2.2) the (I, J )conditions corresponding to I = {1, 3} or I = {2, 3} are also satisfied. Let I = {1}. The equation: a1 = 4 = b11 x1 = 4x1 has the unique solution x1 =1. Since for j =2 or 3, the condition aj > bj 1 x1 is 4 > 3, the (I, J )-conditions corresponding to I = {1} are satisfied. Again by symmetry of (2.2), the (I, J )-conditions are satisfied for I = {2} or I = {3}. This shows that (2.2) satisfies the (I, J )-conditions. In the above example the constant growth rates can be replaced by variable growth rates with averages equal to 4. Also since the (I, J )-conditions are stable under small perturbations of the interaction coefficients and growth rate averages, it is easy to see that there are nonsymmetrical (LV)-systems which satisfy the (I, J )-conditions but none of conditions (2.1). 3. Almost necessary and sufficient conditions for persistence when n = 3 We consider system (LV) when n = 3 and give conditions under which if u(t) = (u1 (t), u2 (t), u3 (t)) is a solution with uk (t0 ) > 0, k = 1–3, then lim sup uk (t) > 0
(3.1)
t→∞
for k = 1–3. These conditions will involve certain inequalities among quantities depending on the averages of the growth rates and the interaction coefficients. These inequalities, which are strict, are sufficient for (3.1) to hold and are also almost necessary in the sense that if any one of them is reversed, then (3.1) does not hold. Throughout this section we assume that if i = j then M[ai ] = bij M[aj ]/bjj .
(3.2)
As our main result will show, systems in which this is not true are structurally unstable in the sense that slight changes in the averages of the growth rates or the interaction coefficients can drastically alter the long-term behavior of solutions. The following theorem concerning the two-species system u1 (t) = u1 (t)[a1 (t) − b11 u1 (t) − b12 u2 (t)], u2 (t) = u2 (t)[a2 (t) − b21 u1 (t) − b22 u2 (t)],
(LV2 )
where the growth rates and interaction coefficients satisfy the assumptions of the previous sections, appeared in [5].
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Theorem 3.1. Let (u1 (t), u2 (t)) be a solution of (LV2 ) such that u1 (t0 ) > 0, u2 (t0 ) > 0. For i = 1, 2, and T > 0 let 1 t0 +T Ai (T ) = ui (t) dt. T t0 (a) If M[a1 ] > b12 M[a2 ]/b22 , M[a2 ] > b21 M[a1 ]/b11 , then for i = 1, 2, lim Ai (T ) = xi∗ ,
T →∞
where x1 = x1∗ , x2 = x2∗ is the unique solution of b11 x1 + b12 x2 = M[a1 ], b21 x1 + b22 x2 = M[a2 ].
(3.3)
(b) If M[a1 ] > b12 M[a2 ]/b22 , M[a2 ] < b21 M[a1 ]/b11 , then lim A1 (T ) = M[a1 ]/b11 ,
T →∞
lim A2 (T ) = 0.
T →∞
(c) If M[a1 ] < b12 M[a2 ]/b22 , M[a2 ] > b21 M[a1 ]/b11 , then lim A1 (T ) = 0,
T →∞
lim A2 (T ) = M[a2 ]/b22 .
T →∞
(d) If M[a1 ] < b12 M[a2 ]/b22 , M[a2 ] < b21 M[a1 ]/b11 , then either (i) A1 (T ) → M[a1 ]/b11 , A2 (T ) → 0 as T → ∞, (ii) A1 (T ) → 0, A2 (T ) → M[a2 ]/b22 as T → ∞, or ∗ (iii) there exists a sequence {Tn }∞ 1 such that Tn+1 > Tn > 0 for n 1, Tn → ∞ as n → ∞ and Ai (Tn ) → xi as ∗ ∗ n → ∞, i = 1, 2, where x1 = x1 , x2 = x2 is the unique solution of (3.3). For proofs of the following two theorems see [5,6]. Theorem 3.2. Consider an (LV) system in which the number of species N = m + r, where m 1, r 1, and assume that the (I, J ) conditions are satisfied in the m-species (LV) system ui (t) = ui (t)[ai (t) −
m
bi u (t)],
1 i m.
=1
Let xl > 0, l = 1, . . . , m, be the unique positive numbers (see [4]) such that for i = 1, . . . , m, M[ai ] =
m
bil xl
l=1
and assume that for j = m + 1, . . . , N, M[aj ] <
m
bj l xl .
l=1
Let A be a compact subset of {(1 , . . . , m )|i > 0, i = 1, . . . , m}. There exists a number d =d(A) > 0 such that if u=col(u1 , . . . , uN ) is any solution of (LV), with (u1 (t0 ), . . . , um (t0 )) ∈ A and 0 < uj (t0 )d(A), j = m + 1, . . . , N, then for j = m + 1, . . . , N, lim uj (t) = 0.
t→∞
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Theorem 3.3. System (LV)3 is persistent if for any permutation of i, j, k of 1,2,3, M[ai ] > bij M[aj ]/bjj , M[aj ] > bj i M[ai ]/bii , M[ak ] > bki xi∗ + bkj xj∗ , where xi = xi∗ , xj = xj∗ is the unique solution of M[ai ] = bii xi + bij xj , M[aj ] = bj i xi + bjj xj . Furthermore, if the first two inequalities hold and the third one is reversed, i.e. M[ak ] < bki xi∗ + bkj xj∗ then the system is not persistent. The above theorem is incomplete in the sense that it corresponds to part (a) of Theorem 3.1, and does not explain what the implications of the remaining parts of this theorem might be. Our next theorem gives a complete classification based on Theorem 3.1. The following result will facilitate the proof of our main theorem. Lemma 3.1. If conditions (d) of Theorem 3.1 hold, M[a3 ] > b31 M[a1 ]/b11 , and M[a3 ] > b32 M[a2 ]/b22 , then M[a3 ] > b31 x1∗ + b32 x2∗ . Proof. We have that 1 = b11
x1∗ x∗ + b12 2 . M[a1 ] M[a1 ]
Since, by (d), b12 b22 > M[a1 ] M[a2 ] we see that 1>
b11 x1∗ b22 x2∗ + . M[a1 ] M[a2 ]
Consequently, by the hypotheses of the lemma b31 b32 ∗ ∗ 1 > x1 + x2 M[a3 ] M[a3 ] and the lemma is proved.
Theorem 3.4. Assume that for ⎡ ⎤ 3 ui (t) = ui (t) ⎣ai (t) − bij uj (t)⎦ ,
i = 1, 2, 3,
(LV3 )
j =1
conditions (3.2) hold for 1i, j 3 and i = j . This system will be persistent if the following conditions are satisfied. Let i, j, k be any permutation of 1, 2, 3. (a) If M[ai ] > bij M[aj ]/bjj and M[aj ] > bj i M[ai ]/bii , then M[ak ] > bki xi∗ + bkj xj∗ ,
(3.4)
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S. Ahmad, I. Stamova / Nonlinear Analysis: Real World Applications 9 (2008) 708 – 717
where xi = xi∗ , xj = xj∗ is the unique solution of M[ai ] = bii xi + bij xj , M[aj ] = bj i xi + bjj xj .
(3.5)
(b) If M[ai ] > bij M[aj ]/bjj , and M[aj ] < bj i M[ai ]/bii , then M[ak ] > bki M[ai ]/bii .
(3.6)
(c) If M[ai ] < bij M[aj ]/bjj and M[aj ] > bj i M[ai ]/bii , then M[ak ] > bkj M[aj ]/bjj .
(3.7)
(d) If M[ai ] < bij M[aj ]/bjj and M[aj ] < bj i M[ai ]/bii , then both of the inequalities (3.6) and (3.7) hold. These conditions are sharp in the following sense: If the conditions of (a) hold and M[ak ] < bki xi∗ + bkj xj∗
(3.8)
then (LV3 ) is not persistent. If the conditions of (b) hold and M[ak ] < bki M[ai ]/bii
(3.9)
then (LV3 ) is not persistent. If the conditions of (c) hold and M[ak ] < bkj M[aj ]/bjj
(3.10)
then (LV3 ) is not persistent. If the conditions of (d) hold and either one of the inequalities (3.9) or (3.10) is satisfied, then (LV3 ) is not persistent. If i, j, k is a fixed permutation of 1,2,3 then (a)–(c) are sharp sufficient conditions for persistence of the kth species, i.e., lim sup uk (t) > 0. t→∞
Proof. If (LV3 ) is not persistent then we have a solution (u#1 (t), u#2 (t), u#3 (t)) of (LV3 ) such that u#j (t0 ) > 0 for j = 1–3 and for some k, 1k 3, u#k (t) → 0 as t → ∞. By permutation of indices we may assume that k = 3. By replacing t0 by a larger number, if necessary, we may assume that a (t) − b3 u#3 (t) > 0 for t t0 − 1 and = 1, 2. If we define for = 1, 2 t t0 , a (t) − b3 u#3 (t), # a (t) = a (t) − (t − t0 + 1)b3 u#3 (t), t0 − 1 t < t0 , t < t0 − 1, a (t), # > 0 and a # < ∞ for = 1, 2. Moreover, since u# (t) → 0 as t → ∞, the then a# is continuous on (−∞, ∞), aL 3 M limit 1 t∗ +T # 1 t∗ +T M[a# ] = lim a (t) dt = lim a (t) dt = M[a ] T →∞ T t∗ T →∞ T t∗
exists uniformly with respect to t∗ ∈ (−∞, ∞) and = 1, 2. For proof of part (a), see [5,6]. Suppose the conditions of part (b) hold. We may assume i = 1. Then M[a1# ] = M[a1 ] > b12
M[a2# ] M[a2 ] = b12 b22 b22
S. Ahmad, I. Stamova / Nonlinear Analysis: Real World Applications 9 (2008) 708 – 717
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and M[a2# ] = M[a2 ] < b21
M[a1# ] M[a1 ] = b21 . b11 b11
Therefore, by part (b) of Theorem 3.1 applied to (LV#2 ), we see that 1 T →∞ T
t0 +T
lim
t0
u#1 (t) dt =
M[a1 ] M[a1# ] = b11 b11
and 1 lim T →∞ T
t0 +T
t0
u#2 (t) dt = 0.
Since (3.6) holds for k = 3, i = 1, and since u#3 (t) → 0 as t → ∞,
1 T →∞ T
t0 +T
lim
(a3 (t) −
t0
3
b3 u# (t)) dt
=1
= M[a3 ] − b31 M[a1 ]/b11 > 0. But since u#3 (t) → 0 as t → ∞, it follows that for T large 1 T
t0 +T
(a3 (t) −
t0
3
b3 u# (t)) dt
=1
u#3 (t0 + T ) 1 = ln <0 T u#3 (t0 )
so we again have a contradiction. This proves part (b). The proof for part (c) is the same as that for part (b) with the roles of u#1 (t) and u#2 (t) interchanged. Suppose that the conditions of part (d) are satisfied. Since M[a1# ] = M[a1 ] < b12 M[a2 ]/b22 = b12 M[a2# ]/b22 and M[a2# ] = M[a2 ] < b21 M[a1 ]/b11 = b21 M[a1# ]/b11 and u1 (t) = u#1 (t), u2 (t) = u#2 (t) is a solution of (LV#2 ) for t t0 , it follows from part (d) of Theorem 3.1 that either t +T lim 1 0 u#1 (t) dt = M[a1 ]/b11 , T →∞ T t0 t +T (3.11) lim T1 t00 u#2 (t) dt = 0, T →∞
or
1 T →∞ T lim 1 T →∞ T
lim
t0 +T t0 t0 +T t0
u#1 (t) dt = 0, u#2 (t) dt = M[a2 ]/b22 ,
(3.12)
or there exists a sequence {Tn }∞ 1 such that Tn+1 > Tn > 0 for n 1, Tn → ∞ as n → ∞ and for = 1, 2 1 n→∞ Tn
t0 +Tn
lim
t0
u# (t) dt = x∗ .
(3.13)
If (3.11) holds, then since, by assumption, M[a3 ] > b31 M[a1 ]/b11 , we get a contradiction just as we did in the proof of part (b). The case in which (3.12) holds also leads to a contradiction as in the proof of part (c).
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S. Ahmad, I. Stamova / Nonlinear Analysis: Real World Applications 9 (2008) 708 – 717
If (3.13) holds, then since u#3 (t) → 0 as t → ∞, it follows from the lemma that t0 +Tn 3 1 # lim a3 (t) − b3 u (t) dt = M[a3 ] − b31 x1∗ − b32 x2∗ > 0. n→∞ Tn t 0 =1
However, for n large t0 +Tn 3 u#3 (t0 + Tn ) 1 1 # a3 (t) − b3 u (t) dt = n < 0, Tn Tn t0 u#3 (t0 ) =1
so we have a contradiction. This proves that the conditions of the theorem are sufficient for persistence of (LV3 ). Case (a), where M[a3 ] < b31 x1∗ + b32 x2∗ , assuming that k = 3, was handled in [5,6]. Suppose that in case (b) i = 1, j = 2, k = 3 and M[a3 ] < b31 M[a1 ]/b11 . Obviously the single species (LV)-system u1 (t) = u1 (t)[a1 (t) − b11 u1 (t)] satisfies the (I, J )-conditions. Since the unique solution x1 = x1∗ of b11 x1 = M[a1 ] is x1∗ = M[a1 ]/b11 and since, for j = 2, 3, M[aj ] < bj 1 M[a1 ]/b11 = bj 1 x1∗ , it follows from Theorem 3.2 (with m = 1) that there are solutions of (LV3 ), u = (u1 , u2 , u3 ), u1 (t0 ) > 0, u2 (t0 ) > 0, u3 (t0 ) > 0 with u2 (t) → 0, u3 (t) → 0 as t → ∞. Therefore (LV3 ) is not persistent. Similarly, if M[a1 ] < b12 M[a2 ]/b22 , M[a3 ] < b32 M[a2 ]/b22 , then there are solutions of (LV3 ) with u (t0 ) > 0, = 1–3 and u1 (t) → 0, u3 (t) → 0 as t → ∞. This together with the above shows that when conditions (b) hold, (3.9) precludes persistence, and when conditions (c) hold, (3.10) precludes persistence. The same argument shows that when conditions (d) hold either (3.9) or (3.10) precludes persistence. The last statement of the theorem is obvious. References [1] S. Ahmad, A.C. Lazer, Necessary and sufficient average growth in a Lotka–Volterra system, Nonlinear Anal. 34 (1998). [2] S. Ahmad, A.C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka–Volterra system, Nonlinear Anal. 40 (2000). [3] S. Ahmad, A.C. Lazer, Average growth and extinction in a competitive Lotka–Volterra system, Nonlinear Anal. 62 (2005). [4] S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006). [5] S. Ahmad, I.M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. Real World Appl. 5 (2004). [6] S. Ahmad, I.M. Stamova, Partial persistence and extinction in N -dimensional competitive systems, Nonlinear Anal. 60 (2005). [7] R. Bellman, Matrix Analysis, second ed., McGraw-Hill, New York, 1975. [8] T. Hallam, L. Svoboda, T. Gard, Persistence and extinction in three species Lotka–Volterra competitive systems, Math. Biosci. 46 (1979). [9] P. Hartman, Ordinary Differential Equations, P. Hartman, Baltimore, 1973. [10] J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, New York, 1988.
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[11] W. Jansen, A permanence theorem for replicator and Lotka–Volterra systems, J. Math. Biol. 25 (1986). [12] R.M. May, W. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975). [13] R. Redheffer, Lotka–Volterra systems with constant interaction coefficients, Nonlinear Anal. 46 (2001). [14] Y. Takeuchi, Global Dynamical Properties of Lotka–Volterra Systems, World Scientific, Singapore, 1996. [15] P. Waltman, A brief survey of persistence in dynamical systems, in: Busenberg, M. Martelli (Eds.), Delay Differential Equations and Dynamical Systems, fifth ed., Springer, Berlin, 1991.