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On the stability of non-autonomous perturbed Lotka–Volterra models
Applied Mathematics and Computation 219 (2013) 6868–6881
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On the stability of non-autonomous perturbed Lotka–Volterra models F. Capone, R. De Luca, S. Rionero ⇑ University of Naples Federico II, Department of Mathematics and Applications ‘R. Caccioppoli’, Complesso Universitario Monte S. Angelo, Via Cinzia, 80126 Naples, Italy
a r t i c l e
i n f o
Keywords: Nonautonomous systems Lotka–Volterra generalized systems Stability/instability via Direct Method
a b s t r a c t The paper is devoted to an extended Lotka–Volterra system of differential equations of predator–prey model. The extension is proposed with perturbation terms, which are null for the positive equilibrium state. In the original Lotka–Volterra system, the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time. The aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state. The reason of the proposed model is that for large time scale, the Lotka–Volterra model is too simple to be realistic. In the paper, the non-autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from Lotka–Volterra model. Theorems are proposed in a renormalized form of the differential equations for time and the two variables. The key point of the paper comes from the use of a Liapunov function introduced in Section 2 which allows to obtain conditions for the asymptotic stability (Section 3) and instability (Section 4) by using a Cetaiev instability theorem following conditions on the renormalized coefficients in time of System (6). An appendix recalls the main results of the Liapunov Direct Method for non-autonomous binary systems of ordinary differential equations. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The celebrated Lotka–Volterra model governing the predation between two species is based, as it well known, on the assumptions: (i) in the absence of predators, the preys increase at a constant rate; (ii) in the absence of preys, the predators decrease at a constant rate; (iii) the rate at which preys are eaten is proportional to the product of the densities of predators and preys. Therefore denoting by a; b; c; d positive constants and by x and y respectively the preys and predators densities, the equations governing the model are of the type:
x_ ¼ xða byÞ;
y_ ¼ yðc þ dxÞ:
⇑ Corresponding author. E-mail addresses: [email protected] (F. Capone), [email protected] (R. De Luca), [email protected] (S. Rionero). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.01.003
ð1Þ
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A well established criticism can be done to (i)–(iii) and hence to (1). In fact, the following remarks hold [1] (1) the growth behavior assumed by (i) is reasonable only for a limited time, since, a continuous increasing of the population will exhaust its resources; (2) the density of each specie does not exhibit any structure (space location, age, differences of sex or genotype, . . .); (3) changes in density are deterministic, ignoring the random effects in the environment influence on the interaction between x and y; (4) the effects of interactions within and between the species are instantaneous, ignoring the influence of delayed processes. In the literature can be found several perturbed Lotka–Volterra models
x_ ¼ xða byÞ þ F;
y_ ¼ yðc þ dxÞ þ G;
ð2Þ
developed by many authors. Different types of perturbation terms F; G have been introduced in order to account for variations of the idealized hypotheses (i)–(iii) as well as to put controls on the growth of both predators and preys (see, for instance [1–17] and for models of binary reaction–diffusion of P.D.Es [18–24] and the references therein). The influence of the perturbation terms on the stability of the positive ecological equilibrium state
c x¼ ; d
a y¼ ; b
ð3Þ
of (1) or on the existence both of periodic solutions or perturbed critical points, have been studied. In particular, in [5–9,14– 17], the perturbation terms are such that
½F ðx;yÞ ¼ ½Gðx;yÞ ¼ 0:
ð4Þ
Our aim here is to consider a general class of non-autonomous perturbation terms verifying (4), in order to study their influence on the stability/instability of (3) and on the existence of periodic solutions. In particular, since in the Lotka–Volterra model (1) (i) ðx; yÞ is stable but not asymptotically stable; (ii) the perturbations ðn ¼ x x; g ¼ y yÞ, with ðn0 – 0; g0 – 0Þ, are periodic in time with the same period; our aim is precisely to introduce, as measure of random effects in the environment that influence the interaction between x and y, the perturbation terms ðF; GÞ guaranteeing (i)’ the asymptotic stability of ðx; yÞ and hence the non-existence of periodic solutions ðx ¼ x þ n; y ¼ y þ gÞ for ðn0 – 0; g0 – 0Þ; (ii)’ the instability of ðx; yÞ; (iii)’ the overcoming of (1)–(3). The non-autonomous perturbation terms considered in the present paper are:
8 < F ¼ f ðtÞxða byÞ þ D1 ðtÞ x y x ; y : G ¼ gðtÞyðc þ dxÞ þ D ðtÞy x y; 2
ð5Þ
x
and hence the perturbed Lotka–Volterra models studied can be written:
8 < x_ ¼ f1 ðtÞða byÞx þ D1 ðtÞ x y x ; y : y_ ¼ f ðtÞðc þ dxÞy þ D ðtÞy x y; 2 2 x
ð6Þ
with f1 ¼ 1 þ f , f2 ¼ 1 þ g. We assume fi : Rþ ! Rþ , (i ¼ 1; 2). In view of (6), one obtains
8 > < x_ ¼ af1 ðtÞ 1 yy x þ xD1 ðtÞ yy xx ; > y_ ¼ cf ðtÞ1 þ xy þ yD ðtÞ x y : : 2 2 x x y
ð7Þ
Setting [18–21]
(
x ¼ xX;
u1 ðtÞ ¼
y ¼ yY; D1 ðtÞ ; af1 ðtÞ
s¼a
u2 ðtÞ ¼
Rt
f ðzÞ dz;
0 1 D2 ðtÞ ; af1 ðtÞ
cf2 ðtÞ wðtÞ ¼ af ; 1 ðtÞ
ð8Þ
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model (7) becomes
(
dX ds dY ds
¼ ð1 YÞX þ u1 ðtÞðY XÞ; ¼ wðtÞð1 þ XÞY þ u2 ðtÞðX YÞ;
ð9Þ
Eq. (9) having ðX ; Y Þ ¼ ð1; 1Þ as critical point where neither species is extinct. We assume that
w; ui 2 L1 ðRþ Þ \ C 1 ðRþ Þ;
ði ¼ 1; 2Þ;
ð10Þ
with
jui ðt1 Þ ui ðt 2 Þj 6 Li jt 1 t 2 j;
ði ¼ 1; 2Þ;
jwðt1 Þ wðt2 Þj 6 Kjt1 t 2 j;
ð11Þ
where t 1 ; t 2 2 Rþ and Li (i ¼ 1; 2), K positive constants. Further we require that
8 > < u1 ðtÞ < 1; > :
8t 2 Rþ :
ð12Þ
wðtÞ þ u2 ðtÞ > 0;
We remark that (i) (10) and (11) guarantee (global) existence and uniqueness of smooth solutions of (9); (ii) (12)1 guarantees that the preys grow up in the absence of predators; (iii) (12)2 guarantees that the predators decrease in the absence of preys. Our aim is to characterize the functions w; ui (i ¼ 1; 2) guaranteeing the nonlinear stability (instability) of the biological meaningful equilibrium state ðX ; Y Þ ¼ ð1; 1Þ, existing 8w; ui (i ¼ 1; 2). The plan of the paper is as follows. Section 2 is devoted to some fundamental preliminaries. In particular a novel Liapunov function is introduced and its time derivative along the solutions of (9) is obtained. In Section 3 conditions guaranteeing the stability of ðX ; Y Þ ¼ ð1; 1Þ are furnished, while the instability is analyzed in Section 4. Final remarks are showed in Section 5. Finally in Appendix A we recall the essential tools of the Direct Method for nonautonomous O.D.Es. 2. Preliminaries In view of (9), it follows that
8 h io R n > < X ¼ X 0 exp 0s 1 YðzÞ þ u1 ðzÞ YðzÞ 1 dz; XðzÞ h io R n > : Y ¼ Y 0 exp 0s wðzÞð1 þ XðzÞÞ þ u2 ðzÞ XðzÞ 1 dz; YðzÞ
ð13Þ
and hence fX 0 > 0; Y 0 > 0g ) fXðsÞ > 0; YðsÞ > 0; 8s > 0g. Setting
X ¼ u þ 1; Y ¼ v þ 1;
ð14Þ
Eq. (9) becomes
(
du ds dv ds
¼ u1 ðtÞu þ ðu1 ðtÞ 1Þv uv ; ¼ ðwðtÞ þ u2 ðtÞÞu u2 ðtÞv þ wðtÞuv :
ð15Þ
In order to study the nonlinear stability/instability of ðX ; Y Þ ¼ ð1; 1Þ we will consider the standard ‘‘energy’’
EðsÞ ¼
1 ðl ðsÞu2 þ l2 ðsÞv 2 Þ; 2 1
ð16Þ
and the Rionero ‘‘energy’’ [18]
VðsÞ ¼
1 fAðu2 þ v 2 Þ þ ½u1 v þ ðw þ u2 Þu2 þ ½ðu1 1Þv þ u2 u2 g; 2
ð17Þ
with
AðtÞ ¼ u1 u2 ðu1 1Þðw þ u2 Þ ¼ wðtÞ½1 u1 ðtÞ þ u2 ðtÞ; and
1
þ
1
þ
li 2 L ðR Þ \ C ðR Þ, (i ¼ 1; 2), to be suitably chosen later.
ð18Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
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Remark 1. We remark that, since w P 0, 8s 2 Rþ and (12)1 holds, then
A P u2 ; and hence u2 P 0 implies A P 0. Further, from (12), it follows that
A ¼ u1 u2 ðu1 1Þðw þ u2 Þ > u1 u2 : Hence if u1 u2 P 0, then A > 0. Along the solutions of (15), it follows that
dE 1 ¼ ds 2
dl1 dl2 2u 1 l 1 u 2 þ 2u2 l2 v 2 þ 2 l1 ðu1 1Þ þ l2 ðw þ u2 Þ uv þ UðsÞ; ds ds
ð19Þ
with
UðsÞ ¼ ðl1 u þ wl2 v Þuv :
ð20Þ
Moreover, setting
8 > < WðsÞ ¼ ðA1 u A3 v Þðuv Þ þ ðA2 v A3 uÞðwuv Þ; I ¼ ðu1 þ u2 Þ; A1 ¼ A þ u22 þ ðw þ u2 Þ2 ; A2 ¼ A þ u21 þ ðu1 1Þ2 ; > : A3 ¼ ½u1 ðw þ u2 Þ þ u2 ðu1 1Þ;
ð21Þ
along the solutions of (15) it turns out that
dV ¼ Pðs; u; v Þ þ Wðs; u; v Þ; ds
ð22Þ
with
8 3 X > > > P ¼ 12 Pi ðs; u; v Þ; > > < i¼1
2 P 1 ¼ 2IA þ dA ðu þ v 2 Þ; ds 2
ð23Þ
d ds d d ¼ ¼ af1 ðtÞ ; dt dt ds ds
ð24Þ
2
du u2 Þ > P2 ¼ dðwþdsu2 Þ u2 þ ds1 v 2 þ 2 d½u1 ðwþ uv ; > ds > > > : du22 2 dðu1 1Þ2 2 d½u2 ðu1 1Þ uv : P 3 ¼ ds u þ ds v þ 2 ds
Remark 2. Since
and af1 ðtÞ > 0, 8t 2 Rþ , then
df df > 0 () > 0; ds dt
8f 2 C 1 ðRþ Þ:
ð25Þ
By virtue of (25) we can state the results for the stability–instability of the null solution of (15) by means of conditions on dE dt and dV instead of dE and dV . dt ds ds Remark 3. Let f : Rþ ! R and
f ¼ inf f; þ R
f ¼ supf ; Rþ
then it turns out that (i) at any instant t 2 Rþ and 8AðtÞ, in any disk of the phase space, centered at the origin O ¼ ð0; 0Þ, there exists a domain in which Vðt; u; v Þ > 0; (ii) if A > 0, there exists a positive constant m1 such that
A ðu2 þ v 2 Þ < V < m1 ðu2 þ v 2 Þ;
8s 2 Rþ ;
ð26Þ
and hence V is limited, has an infinitely small upper limit and is positive definite; (iii) the property (i) holds also for the energy E either when ðl1 Þ > 0 or ðl2 Þ > 0; (iv) the property (ii) holds also for the energy E, when ðli Þ > 0, (i ¼ 1; 2). In fact one has
m2 ðu2 þ v 2 Þ < E < m3 ðu2 þ v 2 Þ;
ð27Þ
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with
m2 <
1 inf ðl1 Þ ; ðl2 Þ ; 2
m3 >
1 sup ðl1 Þ ; ðl2 Þ : 2
ð28Þ
Lemma 1. The polynomial P 2 reduces to
8 < P ¼ dðwþu2 Þ2 ðu v Þ2 2 ds : P ¼ dðwþu2 Þ2 u2 2
for u1 ¼ ðw þ u2 Þ; 8s 2 Rþ ;
ð29Þ
for u1 0; 8s 2 Rþ ;
ds
while P3 reduces to
8 < P ¼ dðu1 1Þ2 ðu v Þ2 3 ds : P ¼ dðu1 1Þ2 v 2 3
for u1 ¼ 1 u2 ; 8s 2 Rþ ;
ð30Þ
for u2 0; 8s 2 Rþ :
ds
If no one of the functions ui ði ¼ 1; 2Þ is identically zero, the polynomials Pi ði ¼ 2; 3Þ, cannot be negative definite. Proof. (29) and (30) are implied by (23). Further, since
8h i2 2 2 du2 u2 ÞÞ u2 Þ > < dðu1 ðwþ ds1 dðwþdsu2 Þ ¼ ddus1 ðw þ u2 Þ u1 dðwþ ; ds ds h i 2 2 > : d½u2 ðu1 1Þ du22 dðu1 1Þ2 ¼ du2 ðu 1Þ u dðu1 1Þ ; ds
ds
ds
ds
1
2
ð31Þ
ds
one immediately deduces that Pi (i ¼ 2; 3), as quadratic forms of u and v, cannot be definite negative when no one of the functions ui (i ¼ 1; 2) is identically zero. h Remark 4. We call critical case, the case in which
AI 0;
8s 2 Rþ :
ð32Þ
Lemma 2. The quadratic polynomial P 2 þ P3 is (i) positive semidefinite either for
d 2 ½u þ ðw þ u2 Þ2 P k1 ; ds 2
u1 ðw þ u2 Þ þ u2 ðu1 1Þ ¼ const:;
or
d 2 ½u þ ðu1 1Þ2 P k2 ; ds 1
8s 2 Rþ ;
ð33Þ
(
u1 u1 1 dðw þ u2 Þ2 du22 ¼ const:; ¼ const:; P k3 ; P k4 ; w þ u2 u2 ds ds
ð34Þ
or
(
u1 w
¼ const:; u2 ¼ 0;
dw2 P k5 ; ds
dðu1 1Þ2 P k6 ; ds
8s 2 Rþ ;
ð35Þ
with ki (i ¼ 1; . . . ; 6) non negative constants; (ii) positive definite if the constants ki appearing – either in (33) or (34) or (35) – are positive; (iii) negative semidefinite either for
(
or
u1 ðw þ u2 Þ þ u2 ðu1 1Þ ¼ const:; d ½u21 þ ðu1 1Þ2 6 k2 ; 8s 2 Rþ ; ds
u22 þ ðu1 1Þ2 6 k1 ;
d ½ ds
ð36Þ
(
u1 u1 1 dðw þ u2 Þ2 du22 ¼ const:; ¼ const:; 6 k3 ; 6 k4 ; w þ u2 u2 ds ds
ð37Þ
or
(
u1 w
¼ const:; u2 ¼ 0;
dw2 6 k5 ; ds
with ki , (i ¼ 1; . . . ; 6) non negative constants;
dðu1 1Þ2 6 k6 ; ds
8s 2 Rþ ;
ð38Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
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(iv) negative definite if the constants ki appearing – either in (36) or (37) or (38) – are positive; (v) undefined in the other cases. Proof. For the proof see [18]. h Remark 5. Apart from the critical case one immediately deduces that (i) if A > 0, 8s 2 Rþ , the existence of a positive constant h such that
P1 6 hðu2 þ v 2 Þ;
ð39Þ
is necessary for guaranteeing the (local) asymptotic stability; (ii) if A < 0, 8s 2 Rþ , the existence of a positive constant h such that
P1 > hðu2 þ v 2 Þ;
ð40Þ
is necessary for guaranteeing the (Cetaev) instability (cfr. Appendix A). For the sake of completeness we recall here some lemmas, proved in [18] that we will use to obtain stability/instability results. Lemma 3. Let
A > 0;
I > 0:
ð41Þ
Then does not exist a positive constant h such that
P1 6 hðu2 þ v 2 Þ;
8s 2 R þ ;
ð42Þ
and P 1 is semidefinite positive for
A P A0 e2I s ;
A0 ¼ Að0Þ;
ð43Þ
and definite positive, according to
P1 P A I ðu2 þ v 2 Þ;
ð44Þ
dA P 0; ds
ð45Þ
for
8s 2 R þ :
Remark 6. We observe that
I ¼ infððu1 þ u2 ÞÞ ¼ supðu1 þ u2 Þ: Hence (41) can be written as
A > 0;
ðu1 þ u2 Þ < 0:
Lemma 4. Let
A > 0;
I < 0:
ð46Þ
Then does not exist a positive constant h such that
P1 P hðu2 þ v 2 Þ;
ð47Þ
and P 1 is semidefinite negative for
A 6 A0 e2I s ;
ð48Þ
and negative definite, either according to
P1 6 A jI jðu2 þ v 2 Þ;
ð49Þ
dA 6 0; ds
ð50Þ
for
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or according to
P1 6 2eA jI jðu2 þ v 2 Þ;
0 < e ¼ const: < 1;
ð51Þ
for
A 6 A0 ð1 eÞe2jI js ;
8s > 0:
ð52Þ
Lemma 5. Let
A < 0;
I < 0:
ð53Þ
Then does not exist a positive constant h such that (42) holds. Further P1 is semidefinite positive for A P A0 e2I s ;
ð54Þ
and positive definite according to
P1 P A I ðu2 þ v 2 Þ;
ð55Þ
when (45) holds.
3. Stability criteria Theorem 1. Let (10)–(12), (46) and (48) or (46) and (50) or (46) and (52) hold together with the condition (iii) or (iv) of Lemma 2. Then the null solution of system (15) is nonlinearly, asymptotically, exponentially (locally) stable. Proof. By virtue of the hypotheses, there exist two positive constants h1 and h2 , such that
(
P1 6 h1 ðu2 þ v 2 Þ;
ð56Þ
P2 þ P3 6 h2 ðu2 þ v 2 Þ; and hence, from (22) and (56), one obtains that
dV h 6 ðu2 þ v 2 Þ þ jWj; ds 2
ð57Þ
with h ¼ infðh1 ; h2 Þð> 0Þ. The boundedness of w; ui (i ¼ 1; 2), implies that
jWj 6 Mðu þ v Þðuv Þ 6
pffiffiffi 3 M 2 2 ðu þ v 2 Þ2 ; 2
ð58Þ
with
M ¼ maxðjA3 þ A2 wj; jA1 þ A3 wjÞ:
ð59Þ
Hence, starting form (57), one obtains: 3 1 dV 6 d1 V þ d2 V 2 ¼ Vðd1 þ d2 V 2 Þ; ds
ð60Þ
with
d1 ¼
h ð> 0Þ; 2m1
d2 ¼
pffiffiffi M 2 3
2A2
ð> 0Þ:
ð61Þ
1
Then the assumption V 20 < dd12 implies, by recursive argument, that:
V 6 V 0 eds ;
1
d ¼ d1 d2 V 20 ð> 0Þ:
ð62Þ
Moreover, by virtue of (26), V and W ¼ ðu2 þ v 2 Þ are equivalent and in particular (119)–(121) are satisfied. Now, since all the hypotheses of the Liapunov (asymptotic) stability theorem are satisfied (cfr. Appendix A, (ii)), the null solution of (15) is nonlinearly, asymptotically, exponentially, locally stable. h Remark 7. By virtue of Lemmas 2–5 and Theorem 1, apart from the critical case IA 0, the conditions (46) or the equivalent conditions
A > 0;
ðAIÞ < 0;
appear to be the basic conditions to guarantee the stability of the null solution of (15).
ð63Þ
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6875
Theorem 2. Let (10)–(12) hold and let (46) hold by virtue of
ðu1 Þ P k1 ;
ðu2 Þ P k2 ;
ð64Þ
with ki ð¼ 1; 2Þ positive constants. Then
(
u1 P 1 ð1 u1 Þs¼0 e2ðk1 eÞs ; w þ u2 6 ðw þ u2 Þs¼0 e2ðk2 eÞs ;
ð65Þ
with 0 6 e < infðk1 ; k2 Þ, guarantee the (local) nonlinear asymptotic exponential stability of the null solution of (15). Proof. For the proof see [18]. h Remark 8. Obviously (15) cannot admit periodic solutions when the conditions guaranteeing the asymptotic stability of the null solution hold. Theorem 3. Let (10)–(12) and (46) hold and let us assume that
8 ðe1 u1 Þ > 0; ðe2 u2 Þ > 0; ðu1 þ u2 Þ > 0; > > > > > wþu wþu > e4ð1e1 ÞFðsÞ ; 8s > 0; < 1u12 < 1u12 s¼0 wþu2 u2 > > > wþ e4ð1e2 ÞGðsÞ ; 8s > 0; > 1u1 1u1 > s¼0 > > R Rs : s FðsÞ ¼ 0 u1 ðzÞ dz; GðsÞ ¼ 0 u2 ðzÞ dz; with
ð66Þ
ei ði ¼ 1; 2Þ constants such that ð1 e1 Þu1 þ ð1 e2 Þu2 > 0;
8s 2 Rþ ;
ð67Þ
then the zero solution of (15) is nonlinearly (locally) asymptotically exponentially stable. Proof. Requiring
(
2u1 l1 þ ddls1 < 2e1 u1 l1 ;
ð68Þ
2u2 l2 þ ddls2 < 2e2 u2 l2 ; one obtains that
(
l1 < l1 ð0Þe2ð1e1 ÞFðsÞ ; l2 < l2 ð0Þe2ð1e2 ÞGðsÞ :
ð69Þ
The hypotheses (66)4 and (66)5, are verified by
l1 ¼
1 w þ u2 2 ; 1 u1
l2 ¼
1 u1 w þ u2
12 :
ð70Þ
With this choice, the energy EðsÞ given by (19) has to satisfy
dE 1 < 2ðe1 u1 Þ l1 u2 þ 2ðe2 u2 Þ l2 v 2 þ U; ds 2
ð71Þ
with U given by (20). Setting
h2 ¼ 2 minððe1 u1 Þ ; ðe2 u2 Þ Þ;
ð72Þ
one obtains
dE < h2 E þ U: ds
ð73Þ
The boundedness of w; ui (i ¼ 1; 2), implies that
pffiffiffi 3 M0 2 2 jUj 6 M ðu þ v Þðuv Þ 6 ðu þ v 2 Þ2 ; 2
ð74Þ
M0 ¼ maxðjl1 j; jwl2 jÞ:
ð75Þ
0
with
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Hence, starting form (73) one obtains: 3 1 dE 6 d01 E þ d02 E2 ¼ Eðd01 þ d02 E2 Þ; ds
ð76Þ
with
d01 ¼
h2 ð> 0Þ; m3
pffiffiffi M0 2
d02 ¼ 1
3
ð> 0Þ:
ð77Þ
2m22
d0
Then the assumption E20 < d10 implies, by recursive argument, that: 2
0
E 6 E0 ed s ;
1
d0 ¼ d01 d02 E20 ð> 0Þ:
ð78Þ
Moreover, by virtue of (27), E and W ¼ ðu2 þ v 2 Þ are equivalent and in particular (119)–(121) are satisfied. Now, since all the hypotheses of the Liapunov (asymptotic) stability theorem are satisfied (cfr. Appendix A, (ii)), the null solution of (15) is nonlinearly, asymptotically, exponentially, locally stable. h Remark 9. We observe that (i) (66)3 is necessary for the consistence of (67), while (67) guarantees the consistence of (66)4 and (66)5; (ii) Theorem 3 does not require necessarily
ðu1 Þ > 0;
ðu2 Þ > 0;
ð79Þ
but can hold also if
u1 u2 < 0; 8s > 0; A > 0:
ð80Þ
In fact, let
(
u1 ¼ u2 ½uðsÞ þ 1; u2 < 0; u > 0; e1 ¼ 12 ; e2 ¼ 12 ; A > 0:
ð81Þ
Then
ð1 e1 Þu1 þ ð1 e2 Þu2 ¼
uþ1 2
3 2
u2 þ u2 > 0;
ð82Þ
is verified by
u > 2; 8s > 0:
ð83Þ
Theorem 4. Let (10)–(12) hold and suppose that (46) hold by virtue of
ðu1 Þ > 0;
ðu2 Þ > 0:
ð84Þ
On assuming that
ð1 u1 Þ ðw þ u2 Þ < ðu1 Þ ðu2 Þ ;
ð85Þ
the zero solution of (15) is nonlinearly (locally) asymptotically exponentially stable. Proof. Choosing
l1
1 ðw þ u2 Þ 2 ¼ ; ð1 u1 Þ
l2 ¼
ð1 u1 Þ ðw þ u2 Þ
12 ;
ð86Þ
it follows that
dE 6 u1 l1 u2 u2 l2 v 2 þ l1 ð1 u1 Þ þ l2 ðw þ u2 Þ uv þ U: ds
ð87Þ
In view of (85) and (86) one obtains
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dE 6 ðu1 Þ l1 u2 ðu2 Þ l2 v 2 þ 2 ð1 u1 Þ ðw þ u2 Þ uv þ U: ds
ð88Þ
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Since
6877
l1 ¼ l1 2 it turns out that 8 > < ð1 u1 Þ ðw þ u2 Þ < ðu1 Þ ðu2 Þ ¼ l1 ðu1 Þ l2 ðu2 Þ ; ð1 u1 Þ ðw þ u2 Þ ¼ g2 l1 ðu1 Þ l2 ðu2 Þ ; > : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 u1 Þ ðw þ u2 Þ uv 6 gðl1 ðu1 Þ u2 þ l2 ðu2 Þ v 2 Þ;
ð89Þ
with 0 < g ¼ const: < 1. Then (88) becomes
dE 6 ð1 gÞ ðu1 Þ l1 u2 þ l2 ðu2 Þ v 2 þ U; ds
ð90Þ
and hence
dE 6 h3 E þ U; ds
ð91Þ
h3 ¼ 2ð1 gÞ minððu1 Þ ; ðu2 Þ Þ:
ð92Þ
with
Following the same procedure used in Theorem 3, the thesis is hold. h Theorem 5. Let (10)–(12) hold together with (46) by virtue of
ðu1 Þ > 0;
ðu2 Þ > 0;
ð93Þ
and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 u1 Þ þ ðw þ u2 Þ < 2 ðu1 Þ ðu2 Þ :
ð94Þ
Then the null solution of system (15) is nonlinearly (locally) asymptotically exponentially stable. Proof. Choosing
l1 ¼ l2 ¼ 1, from (19) it follows that
dE 1 ¼ 2u1 u2 2u2 v 2 þ 2ðu1 1 þ u2 þ wÞuv þ U ds 2 6 ðu1 Þ u2 ðu2 Þ v 2 þ ðð1 u1 Þ þ ðw þ u2 Þ Þuv þ U:
ð95Þ
From (94), 9g ¼ const: 20; 1½ such that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 u1 Þ þ ðw þ u2 Þ ¼ 2g ðu1 Þ ðu2 Þ :
ð96Þ
Hence
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dE 6 ðu1 Þ u2 ðu2 Þ v 2 þ 2g ðu1 Þ ðu2 Þ uv þ U 6 h4 ð1 gÞðu2 þ v 2 Þ þ U; ds
ð97Þ
with h4 ¼ minððu1 Þ ; ðu2 Þ Þ. Then following the same procedure used in Theorem 3, the thesis is hold. h Theorem 6. Let (10)–(12) hold. Let us suppose that
A > 0;
I 0;
ð98Þ
together with (iii) or (iv) of Lemma 2. Then if
dA 6 0 8s 2 Rþ ; ds
ð99Þ
or if
dA ~ ¼ k; ds
~ ¼ const: > 0; k
then the null solution of (15) is nonlinearly (locally) asymptotically, exponentially, stable.
ð100Þ
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Proof. From (17), by virtue of (98), one immediately obtains that V is positive definite. Moreover, from (22), on taking into account (ii) or (iv) of Lemma 2, it follows that
dV 6 hðu2 þ v 2 Þ þ jWj: ds Adopting the same procedure followed in Theorem 1, the thesis is hold.
h
Remark 10. When u1 ¼ u2 ¼ 0, (9) becomes
(
dX ds dY ds
¼ ð1 YÞX; ¼ wð1 þ XÞY;
that is a nonautonomous Lotka–Volterra model which has been analyzed by Rionero [21].
4. Instability criteria Instability criteria can be obtained, of course, by means either of the Liapunov function (17) or the function (16). We here recall – for the sake of completeness – the instability theorem obtained in [21] by the function (17) (cfr. Theorems 7–10) and concentrated ourselves on the instability theorems obtained by using the function (16). Theorem 7. Let (10)–(12), (41), (45) hold together with (i) of Lemma 2. Then the null solution of (15) is unstable. Theorem 8. Let (10)–(12), (53), (45) and (i) of Lemma 2 hold. Then the null solution of (15) is unstable. Theorem 9. Let
I 0;
ð101Þ
and (i) of Lemma 2 hold. Then if
dA ~ ¼ const > 0; Pk ds
ð102Þ
the null solution of (15) is unstable. Theorem 10. Let (10)–(12) hold. If
A 0;
ð103Þ
and (ii) of Lemma 2 hold, then the null solution of (15) is (Cetaev) unstable. Theorem 11. Let (41) hold by virtue of
u1 6 h1 ; u2 6 h2 ;
ð104Þ
with hi (i ¼ 1; 2) positive constants. Then
(
w þ u2 > ðw þ u2 Þs¼0 e2ðh1 eÞs ;
u1 < 1 ð1 u1 Þs¼0 e2ðh2 eÞs ;
ð105Þ
with 0 < e < infðh1 ; h2 Þ guarantee the instability of the null solution of (15). Proof. Choosing
l1 ¼ w þ u2 ; l2 ¼ 1 u1 ;
ð106Þ
it follows that
E¼
1 ðw þ u2 Þu2 þ ð1 u1 Þv 2 ; 2
ð107Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
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is positive definite and
dE 1 ¼ ds 2
dðw þ u2 Þ dð1 u1 Þ 2u1 ðw þ u2 Þ u2 þ 2u2 ð1 u1 Þ v 2 ; ds ds
ð108Þ
where we have disregarded the contribution of U. Then (12)2 and (105) guarantee that
( dðwþu
2Þ 2 ds dð1u1 Þ 2 ds
u1 ðw þ u2 Þ > 2eðw þ u2 Þ > 0; u2 ð1 u1 Þ > 2eð1 u1 Þ > 0:
ð109Þ
Hence all the hypotheses of the instability Liapunov theorem (cfr. Appendix A) are verified. h Theorem 12. Let (10)–(12) and (53) hold by virtue of
ju 2 j 6 h2 ;
ðu1 Þ P h1 ;
ð110Þ
with hi (i ¼ 1; 2) positive constants such that h1 > h2 . Then
(
w þ u2 > ðw þ u2 Þ0 e2ðeþh2 Þs ;
u1 > 1 þ ðu1 1Þ0 e2ðeh1 Þs ;
ð111Þ
imply the instability of the null solution of (15). Proof. Since (53) hold, we have to require
u1 u2 < ðu1 1Þðw þ u2 Þ þ A ; 8s 2 Rþ :
ð112Þ
In fact, if h1 > h2 it follows that
u1 þ u2 P h1 ju2 j P h1 h2 > 0; and hence I < 0. Choosing
l1 ¼ u1 1; l2 ¼ w þ u2 ;
ð113Þ
one has
E¼
1 ðu1 1Þu2 þ ðw þ u2 Þv 2 ; 2
ð114Þ
and
dE 1 ¼ ds 2
h i du1 dðw þ u2 Þ 2u1 ðu1 1Þ u2 þ 2u2 ðw þ u2 Þ v 2 þ 2 ðu1 1Þ2 þ ðw þ u2 Þ2 uv ; ds ds
ð115Þ
where we have disregarded the contribution of U. Hence
dE 1 P ds 2
dðu1 1Þ dðw þ u2 Þ 2u1 ðu1 1Þ u2 þ 2u2 ðw þ u2 Þ v 2 ; ds ds
ð116Þ
and the conditions (111) guarantee that
( dðu
u1 ðu1 1Þ > 2eðu1 1Þ > 0; u2 ðw þ u2 Þ > 2eðw þ u2 Þ > 0:
1 1Þ 2 ds dðwþu2 Þ 2 ds
ð117Þ
Hence E satisfies all the hypotheses of the Cetaev instability theorem (Appendix A). h
5. Final remarks (i) A class of generalized non-autonomous bidimensional Lotka–Volterra model of binary O.D.Es is introduced; (ii) The existence of biological meaningful equilibrium state is guaranteed together with the growing up of the preys in the absence of predators and predators decreasing in the absence of preys; (iii) The nonlinear stability of the equilibrium state is studied.
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F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
Acknowledgments This paper has been performed under the auspices of the G.N.F.M. of I.N.d.A.M. and Programma F.A.R.O. (Finanziamenti per l’ Avvio di Ricerche Originali, III tornata) ‘‘Controllo e stabilità di processi diffusivi nell’ambiente’’, Polo delle Scienze e Tecnologie, Università degli Studi di Napoli Federico II. Appendix A. Essential ingredients of the Liapunov Direct Method for non autonomous binary systems of O.D.Es We recall here the essential ingredients of the Liapunov Direct Method for nonautonomous binary systems of O.D.Es. (cfr [25, pp. 221–228]). Let Vðx; y; tÞ be a real single valued function – depending explicitly on t – defined in the space–time domain
Dðt0 ; rÞ : ðt; x; yÞ 2 D ft P t 0 ; x2 þ y2 6 rg;
ð118Þ
where t0 and r are constants such that ðt0 P 0; r > 0Þ. If – for a sufficiently large t 0 and a sufficiently small r – exists (an independent of t) function Wðx; yÞ, positive in Dðt0 ; rÞ, such that
Vð0; 0; tÞ ¼ 0;
8t P t 0 ;
Vðx; y; tÞ P Wðx; yÞ;
ð119Þ
ðV P Wðx; yÞÞ;
then V is said to be positive (negative) definite. V is said to be positive (negative) semidefinite if has only positive (negative) sign in D but can become zero at some point other than the origin. A function Vðx; y; tÞ said to admit an infinitely small upper limit if exists a t0 P 0 and r > 0 such that
jVj < Wðx; yÞ in Dðt 0 ; lÞ; 2
ð120Þ 2
with W definite in the disk x þ y 6 r and such that
lim Wðx; yÞ ¼ 0:
ð121Þ
x2 þy2 !0
The main stability theorems of the Direct Method for non autonomous systems can be summarized as follows Liapunov stability theorems: The existence of a positive function V implies (i) stability of the null solution if the time derivative of V along the solutions is negative semidefinite; (ii) asymptotic stability if V admits an upper bound which is infinitely small at the origin and V_ is negative definite. The main instability theorems for nonautonomous systems (cfr [25, pp. 226–227]) are Liapunov instability theorem: If exists a function V such that has an infinitely small upper limit and its time derivative V_ along the solutions is definite and also if for t P t 0 (with arbitrarily large t0 ) the function V can have the same sign as V_ in a neighborhood of the origin, then the null solution is unstable. Cetaev instability theorem: If for any disk D of the phase space centered at the origin, there exist a function V and an open subset H D, such that 8t P t 0 (i) (ii) (iii) (iv)
V is bounded in H and assumes positive values in H \ D, V vanishes on @ H \ D, the time derivative of V, along the solution, is positive definite in H, 0 2 @ H,
then the null solution is unstable. References [1] H.N. Comins, D.W.E. Blatt, Prey predator models in spatially heterogeneous environments, J. Theor. Biol. 48 (1974) 75–83. [2] R. Dutt, P.K. Glosh, Nonlinear correction to Lotka Volterra oscillations in a prey predator system, Math. Biosci. 27 (1975) 9–16. [3] R. Dutt, P.K. Glosh, B.B. Karmaker, Application of perturbation theory to the nonlinear Volterra Gause Witt model for prey predator interaction, Bull. Math. Biol. 37 (1975) 139–146. [4] H.I. Freedman, P. Waltman, Perturbation of two dimensional predator prey equations, SIAM J. Appl. Math. 28 (1975) 1–10. [5] H.I. Freedman, P. Waltman, Periodic solutions of perturbed Lotka Volterra systems, in: H.A. Antosiewicz (Ed.), International Conference on Differential Equations, Academic Press, New York, 1975, pp. 312–316. [6] H.I. Freedman, P. Waltman, Perturbation of two dimensional predator prey equations with an unperturbed critical point, SIAM J. Appl. Math. 29 (1975) 719–733. [7] P. Fergola, A.H. Nayfeh, Application of the method of multiple scales to perturbed Lotka Volterra models, in: S. Giambó (Ed.), Proceedings of Vth International Conference ‘‘Waves and Stability in Continuous Media’’, Editel, Cosenza, 1987. [8] P. Fergola, S. Rionero, C. Tenneriello, Asymptotic stability of a perturbed Lotka Volterra system, in: Proceedings of Vth International Conference ‘‘Waves and Stability in Continuous Media’’, Ser. Adv. Math. Appl. Sci. 4 (1990).
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[9] P. Fergola, S. Rionero, C. Tenneriello, On the stability of Lotka Volterra perturbed equations, Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat. 48 (1991) 235–256. [10] H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker Inc., New York, 1980. [11] S.A. Levin, Dispersion and population interactions, Am. Nat. 108 (1974) 207–228. [12] P.A. Samuelson, Generalized predator prey oscillations in ecological and economic equilibrium, Proc. Nat. Acad. Sci. USA 68 (1971) 980–983. [13] C. Walter, The global asymptotic stability of prey predator systems with second order dissipation, Bull. Math. Biol. 36 (1974) 215–217. [14] F. Capone, S. Rionero, Attractivity conditions for a perturbed Lotka Volterra model, in: Proceedings of VIth International Conference ‘‘Waves and Stability in Continuous Media’’, Le Matematiche 46 (1991) 55–63. [15] R. De Luca, Nonlinear stability for a class of generalized Lotka–Volterra models, Rend. Accad. Sci. Fis. Mat. Napoli LXXVII (4) (2010) 117–132. [16] R. De Luca, On the long-time dynamics of nonautonomous predator–prey models with mutual interference, Ric. Mat. 61 (2) (2012) 275–290. [17] R. De Luca, On the asymptotic stability of an Hassell predator–prey model with mutual interference, Acta Appl. Math. 122 (1) (2012) 191–204. [18] S. Rionero, Stability–instability criteria for non-autonomous binary systems of O.D.Es and P.D.Es., Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. Rend. Lincei (9) Mat. Appl. 20 (2009) 1–21. [19] S. Rionero, L2 stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.Es., Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. Rend. Lincei (9) Mat. Appl. 16 (2005) 227–238. [20] S. Rionero, A rigorous reduction of the stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.Es. to the stability of the solutions to a linear binary system of O.D.Es., J. Math. Anal. Appl. 319 (2006) 377–397. [21] S. Rionero, On the nonlinear stability of nonautonomous binary systems, Nonlinear Anal. 75 (2012) 2338–2348. [22] F. Capone, M. Piedisacco, S. Rionero, Nonlinear stability for reaction- diffusion Lotka–Volterra model with Beddington–DeAngelis functional response, Rend. Accad. Sci. Fis. Mat. Napoli LXXIII (4) (2006) 85–97. [23] F. Capone, Nonlinear stability for reaction-diffusion Lotka–Volterra predator–prey model equations with Holling Type 2 functional response, Mathematical Physics Models and Engineering Sciences Societá Nazionale di Scienze, Lettere e Arti in Napoli. Memorie dell’Accademia di Scienze Fisiche e Matematiche, 2008, pp. 73–84. ISBN-13 978-88-207-4057-3. [24] F. Capone, On the dynamics of predator–prey models with the Beddington–DeAngelis functional response, under Robin boundary conditions, Ric. Mat. 57 (2008) 137–157. [25] D.R. Merkin, Introduction to the Theory of Stability, Texts in Applied Mathematics, vol. 24, Springer, 1997.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On the stability of non-autonomous perturbed Lotka–Volterra models F. Capone, R. De Luca, S. Rionero ⇑ University of Naples Federico II, Department of Mathematics and Applications ‘R. Caccioppoli’, Complesso Universitario Monte S. Angelo, Via Cinzia, 80126 Naples, Italy
a r t i c l e
i n f o
Keywords: Nonautonomous systems Lotka–Volterra generalized systems Stability/instability via Direct Method
a b s t r a c t The paper is devoted to an extended Lotka–Volterra system of differential equations of predator–prey model. The extension is proposed with perturbation terms, which are null for the positive equilibrium state. In the original Lotka–Volterra system, the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time. The aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state. The reason of the proposed model is that for large time scale, the Lotka–Volterra model is too simple to be realistic. In the paper, the non-autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from Lotka–Volterra model. Theorems are proposed in a renormalized form of the differential equations for time and the two variables. The key point of the paper comes from the use of a Liapunov function introduced in Section 2 which allows to obtain conditions for the asymptotic stability (Section 3) and instability (Section 4) by using a Cetaiev instability theorem following conditions on the renormalized coefficients in time of System (6). An appendix recalls the main results of the Liapunov Direct Method for non-autonomous binary systems of ordinary differential equations. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The celebrated Lotka–Volterra model governing the predation between two species is based, as it well known, on the assumptions: (i) in the absence of predators, the preys increase at a constant rate; (ii) in the absence of preys, the predators decrease at a constant rate; (iii) the rate at which preys are eaten is proportional to the product of the densities of predators and preys. Therefore denoting by a; b; c; d positive constants and by x and y respectively the preys and predators densities, the equations governing the model are of the type:
x_ ¼ xða byÞ;
y_ ¼ yðc þ dxÞ:
⇑ Corresponding author. E-mail addresses: [email protected] (F. Capone), [email protected] (R. De Luca), [email protected] (S. Rionero). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.01.003
ð1Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
6869
A well established criticism can be done to (i)–(iii) and hence to (1). In fact, the following remarks hold [1] (1) the growth behavior assumed by (i) is reasonable only for a limited time, since, a continuous increasing of the population will exhaust its resources; (2) the density of each specie does not exhibit any structure (space location, age, differences of sex or genotype, . . .); (3) changes in density are deterministic, ignoring the random effects in the environment influence on the interaction between x and y; (4) the effects of interactions within and between the species are instantaneous, ignoring the influence of delayed processes. In the literature can be found several perturbed Lotka–Volterra models
x_ ¼ xða byÞ þ F;
y_ ¼ yðc þ dxÞ þ G;
ð2Þ
developed by many authors. Different types of perturbation terms F; G have been introduced in order to account for variations of the idealized hypotheses (i)–(iii) as well as to put controls on the growth of both predators and preys (see, for instance [1–17] and for models of binary reaction–diffusion of P.D.Es [18–24] and the references therein). The influence of the perturbation terms on the stability of the positive ecological equilibrium state
c x¼ ; d
a y¼ ; b
ð3Þ
of (1) or on the existence both of periodic solutions or perturbed critical points, have been studied. In particular, in [5–9,14– 17], the perturbation terms are such that
½F ðx;yÞ ¼ ½Gðx;yÞ ¼ 0:
ð4Þ
Our aim here is to consider a general class of non-autonomous perturbation terms verifying (4), in order to study their influence on the stability/instability of (3) and on the existence of periodic solutions. In particular, since in the Lotka–Volterra model (1) (i) ðx; yÞ is stable but not asymptotically stable; (ii) the perturbations ðn ¼ x x; g ¼ y yÞ, with ðn0 – 0; g0 – 0Þ, are periodic in time with the same period; our aim is precisely to introduce, as measure of random effects in the environment that influence the interaction between x and y, the perturbation terms ðF; GÞ guaranteeing (i)’ the asymptotic stability of ðx; yÞ and hence the non-existence of periodic solutions ðx ¼ x þ n; y ¼ y þ gÞ for ðn0 – 0; g0 – 0Þ; (ii)’ the instability of ðx; yÞ; (iii)’ the overcoming of (1)–(3). The non-autonomous perturbation terms considered in the present paper are:
8 < F ¼ f ðtÞxða byÞ þ D1 ðtÞ x y x ; y : G ¼ gðtÞyðc þ dxÞ þ D ðtÞy x y; 2
ð5Þ
x
and hence the perturbed Lotka–Volterra models studied can be written:
8 < x_ ¼ f1 ðtÞða byÞx þ D1 ðtÞ x y x ; y : y_ ¼ f ðtÞðc þ dxÞy þ D ðtÞy x y; 2 2 x
ð6Þ
with f1 ¼ 1 þ f , f2 ¼ 1 þ g. We assume fi : Rþ ! Rþ , (i ¼ 1; 2). In view of (6), one obtains
8 > < x_ ¼ af1 ðtÞ 1 yy x þ xD1 ðtÞ yy xx ; > y_ ¼ cf ðtÞ1 þ xy þ yD ðtÞ x y : : 2 2 x x y
ð7Þ
Setting [18–21]
(
x ¼ xX;
u1 ðtÞ ¼
y ¼ yY; D1 ðtÞ ; af1 ðtÞ
s¼a
u2 ðtÞ ¼
Rt
f ðzÞ dz;
0 1 D2 ðtÞ ; af1 ðtÞ
cf2 ðtÞ wðtÞ ¼ af ; 1 ðtÞ
ð8Þ
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model (7) becomes
(
dX ds dY ds
¼ ð1 YÞX þ u1 ðtÞðY XÞ; ¼ wðtÞð1 þ XÞY þ u2 ðtÞðX YÞ;
ð9Þ
Eq. (9) having ðX ; Y Þ ¼ ð1; 1Þ as critical point where neither species is extinct. We assume that
w; ui 2 L1 ðRþ Þ \ C 1 ðRþ Þ;
ði ¼ 1; 2Þ;
ð10Þ
with
jui ðt1 Þ ui ðt 2 Þj 6 Li jt 1 t 2 j;
ði ¼ 1; 2Þ;
jwðt1 Þ wðt2 Þj 6 Kjt1 t 2 j;
ð11Þ
where t 1 ; t 2 2 Rþ and Li (i ¼ 1; 2), K positive constants. Further we require that
8 > < u1 ðtÞ < 1; > :
8t 2 Rþ :
ð12Þ
wðtÞ þ u2 ðtÞ > 0;
We remark that (i) (10) and (11) guarantee (global) existence and uniqueness of smooth solutions of (9); (ii) (12)1 guarantees that the preys grow up in the absence of predators; (iii) (12)2 guarantees that the predators decrease in the absence of preys. Our aim is to characterize the functions w; ui (i ¼ 1; 2) guaranteeing the nonlinear stability (instability) of the biological meaningful equilibrium state ðX ; Y Þ ¼ ð1; 1Þ, existing 8w; ui (i ¼ 1; 2). The plan of the paper is as follows. Section 2 is devoted to some fundamental preliminaries. In particular a novel Liapunov function is introduced and its time derivative along the solutions of (9) is obtained. In Section 3 conditions guaranteeing the stability of ðX ; Y Þ ¼ ð1; 1Þ are furnished, while the instability is analyzed in Section 4. Final remarks are showed in Section 5. Finally in Appendix A we recall the essential tools of the Direct Method for nonautonomous O.D.Es. 2. Preliminaries In view of (9), it follows that
8 h io R n > < X ¼ X 0 exp 0s 1 YðzÞ þ u1 ðzÞ YðzÞ 1 dz; XðzÞ h io R n > : Y ¼ Y 0 exp 0s wðzÞð1 þ XðzÞÞ þ u2 ðzÞ XðzÞ 1 dz; YðzÞ
ð13Þ
and hence fX 0 > 0; Y 0 > 0g ) fXðsÞ > 0; YðsÞ > 0; 8s > 0g. Setting
X ¼ u þ 1; Y ¼ v þ 1;
ð14Þ
Eq. (9) becomes
(
du ds dv ds
¼ u1 ðtÞu þ ðu1 ðtÞ 1Þv uv ; ¼ ðwðtÞ þ u2 ðtÞÞu u2 ðtÞv þ wðtÞuv :
ð15Þ
In order to study the nonlinear stability/instability of ðX ; Y Þ ¼ ð1; 1Þ we will consider the standard ‘‘energy’’
EðsÞ ¼
1 ðl ðsÞu2 þ l2 ðsÞv 2 Þ; 2 1
ð16Þ
and the Rionero ‘‘energy’’ [18]
VðsÞ ¼
1 fAðu2 þ v 2 Þ þ ½u1 v þ ðw þ u2 Þu2 þ ½ðu1 1Þv þ u2 u2 g; 2
ð17Þ
with
AðtÞ ¼ u1 u2 ðu1 1Þðw þ u2 Þ ¼ wðtÞ½1 u1 ðtÞ þ u2 ðtÞ; and
1
þ
1
þ
li 2 L ðR Þ \ C ðR Þ, (i ¼ 1; 2), to be suitably chosen later.
ð18Þ
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Remark 1. We remark that, since w P 0, 8s 2 Rþ and (12)1 holds, then
A P u2 ; and hence u2 P 0 implies A P 0. Further, from (12), it follows that
A ¼ u1 u2 ðu1 1Þðw þ u2 Þ > u1 u2 : Hence if u1 u2 P 0, then A > 0. Along the solutions of (15), it follows that
dE 1 ¼ ds 2
dl1 dl2 2u 1 l 1 u 2 þ 2u2 l2 v 2 þ 2 l1 ðu1 1Þ þ l2 ðw þ u2 Þ uv þ UðsÞ; ds ds
ð19Þ
with
UðsÞ ¼ ðl1 u þ wl2 v Þuv :
ð20Þ
Moreover, setting
8 > < WðsÞ ¼ ðA1 u A3 v Þðuv Þ þ ðA2 v A3 uÞðwuv Þ; I ¼ ðu1 þ u2 Þ; A1 ¼ A þ u22 þ ðw þ u2 Þ2 ; A2 ¼ A þ u21 þ ðu1 1Þ2 ; > : A3 ¼ ½u1 ðw þ u2 Þ þ u2 ðu1 1Þ;
ð21Þ
along the solutions of (15) it turns out that
dV ¼ Pðs; u; v Þ þ Wðs; u; v Þ; ds
ð22Þ
with
8 3 X > > > P ¼ 12 Pi ðs; u; v Þ; > > < i¼1
2 P 1 ¼ 2IA þ dA ðu þ v 2 Þ; ds 2
ð23Þ
d ds d d ¼ ¼ af1 ðtÞ ; dt dt ds ds
ð24Þ
2
du u2 Þ > P2 ¼ dðwþdsu2 Þ u2 þ ds1 v 2 þ 2 d½u1 ðwþ uv ; > ds > > > : du22 2 dðu1 1Þ2 2 d½u2 ðu1 1Þ uv : P 3 ¼ ds u þ ds v þ 2 ds
Remark 2. Since
and af1 ðtÞ > 0, 8t 2 Rþ , then
df df > 0 () > 0; ds dt
8f 2 C 1 ðRþ Þ:
ð25Þ
By virtue of (25) we can state the results for the stability–instability of the null solution of (15) by means of conditions on dE dt and dV instead of dE and dV . dt ds ds Remark 3. Let f : Rþ ! R and
f ¼ inf f; þ R
f ¼ supf ; Rþ
then it turns out that (i) at any instant t 2 Rþ and 8AðtÞ, in any disk of the phase space, centered at the origin O ¼ ð0; 0Þ, there exists a domain in which Vðt; u; v Þ > 0; (ii) if A > 0, there exists a positive constant m1 such that
A ðu2 þ v 2 Þ < V < m1 ðu2 þ v 2 Þ;
8s 2 Rþ ;
ð26Þ
and hence V is limited, has an infinitely small upper limit and is positive definite; (iii) the property (i) holds also for the energy E either when ðl1 Þ > 0 or ðl2 Þ > 0; (iv) the property (ii) holds also for the energy E, when ðli Þ > 0, (i ¼ 1; 2). In fact one has
m2 ðu2 þ v 2 Þ < E < m3 ðu2 þ v 2 Þ;
ð27Þ
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with
m2 <
1 inf ðl1 Þ ; ðl2 Þ ; 2
m3 >
1 sup ðl1 Þ ; ðl2 Þ : 2
ð28Þ
Lemma 1. The polynomial P 2 reduces to
8 < P ¼ dðwþu2 Þ2 ðu v Þ2 2 ds : P ¼ dðwþu2 Þ2 u2 2
for u1 ¼ ðw þ u2 Þ; 8s 2 Rþ ;
ð29Þ
for u1 0; 8s 2 Rþ ;
ds
while P3 reduces to
8 < P ¼ dðu1 1Þ2 ðu v Þ2 3 ds : P ¼ dðu1 1Þ2 v 2 3
for u1 ¼ 1 u2 ; 8s 2 Rþ ;
ð30Þ
for u2 0; 8s 2 Rþ :
ds
If no one of the functions ui ði ¼ 1; 2Þ is identically zero, the polynomials Pi ði ¼ 2; 3Þ, cannot be negative definite. Proof. (29) and (30) are implied by (23). Further, since
8h i2 2 2 du2 u2 ÞÞ u2 Þ > < dðu1 ðwþ ds1 dðwþdsu2 Þ ¼ ddus1 ðw þ u2 Þ u1 dðwþ ; ds ds h i 2 2 > : d½u2 ðu1 1Þ du22 dðu1 1Þ2 ¼ du2 ðu 1Þ u dðu1 1Þ ; ds
ds
ds
ds
1
2
ð31Þ
ds
one immediately deduces that Pi (i ¼ 2; 3), as quadratic forms of u and v, cannot be definite negative when no one of the functions ui (i ¼ 1; 2) is identically zero. h Remark 4. We call critical case, the case in which
AI 0;
8s 2 Rþ :
ð32Þ
Lemma 2. The quadratic polynomial P 2 þ P3 is (i) positive semidefinite either for
d 2 ½u þ ðw þ u2 Þ2 P k1 ; ds 2
u1 ðw þ u2 Þ þ u2 ðu1 1Þ ¼ const:;
or
d 2 ½u þ ðu1 1Þ2 P k2 ; ds 1
8s 2 Rþ ;
ð33Þ
(
u1 u1 1 dðw þ u2 Þ2 du22 ¼ const:; ¼ const:; P k3 ; P k4 ; w þ u2 u2 ds ds
ð34Þ
or
(
u1 w
¼ const:; u2 ¼ 0;
dw2 P k5 ; ds
dðu1 1Þ2 P k6 ; ds
8s 2 Rþ ;
ð35Þ
with ki (i ¼ 1; . . . ; 6) non negative constants; (ii) positive definite if the constants ki appearing – either in (33) or (34) or (35) – are positive; (iii) negative semidefinite either for
(
or
u1 ðw þ u2 Þ þ u2 ðu1 1Þ ¼ const:; d ½u21 þ ðu1 1Þ2 6 k2 ; 8s 2 Rþ ; ds
u22 þ ðu1 1Þ2 6 k1 ;
d ½ ds
ð36Þ
(
u1 u1 1 dðw þ u2 Þ2 du22 ¼ const:; ¼ const:; 6 k3 ; 6 k4 ; w þ u2 u2 ds ds
ð37Þ
or
(
u1 w
¼ const:; u2 ¼ 0;
dw2 6 k5 ; ds
with ki , (i ¼ 1; . . . ; 6) non negative constants;
dðu1 1Þ2 6 k6 ; ds
8s 2 Rþ ;
ð38Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
6873
(iv) negative definite if the constants ki appearing – either in (36) or (37) or (38) – are positive; (v) undefined in the other cases. Proof. For the proof see [18]. h Remark 5. Apart from the critical case one immediately deduces that (i) if A > 0, 8s 2 Rþ , the existence of a positive constant h such that
P1 6 hðu2 þ v 2 Þ;
ð39Þ
is necessary for guaranteeing the (local) asymptotic stability; (ii) if A < 0, 8s 2 Rþ , the existence of a positive constant h such that
P1 > hðu2 þ v 2 Þ;
ð40Þ
is necessary for guaranteeing the (Cetaev) instability (cfr. Appendix A). For the sake of completeness we recall here some lemmas, proved in [18] that we will use to obtain stability/instability results. Lemma 3. Let
A > 0;
I > 0:
ð41Þ
Then does not exist a positive constant h such that
P1 6 hðu2 þ v 2 Þ;
8s 2 R þ ;
ð42Þ
and P 1 is semidefinite positive for
A P A0 e2I s ;
A0 ¼ Að0Þ;
ð43Þ
and definite positive, according to
P1 P A I ðu2 þ v 2 Þ;
ð44Þ
dA P 0; ds
ð45Þ
for
8s 2 R þ :
Remark 6. We observe that
I ¼ infððu1 þ u2 ÞÞ ¼ supðu1 þ u2 Þ: Hence (41) can be written as
A > 0;
ðu1 þ u2 Þ < 0:
Lemma 4. Let
A > 0;
I < 0:
ð46Þ
Then does not exist a positive constant h such that
P1 P hðu2 þ v 2 Þ;
ð47Þ
and P 1 is semidefinite negative for
A 6 A0 e2I s ;
ð48Þ
and negative definite, either according to
P1 6 A jI jðu2 þ v 2 Þ;
ð49Þ
dA 6 0; ds
ð50Þ
for
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F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
or according to
P1 6 2eA jI jðu2 þ v 2 Þ;
0 < e ¼ const: < 1;
ð51Þ
for
A 6 A0 ð1 eÞe2jI js ;
8s > 0:
ð52Þ
Lemma 5. Let
A < 0;
I < 0:
ð53Þ
Then does not exist a positive constant h such that (42) holds. Further P1 is semidefinite positive for A P A0 e2I s ;
ð54Þ
and positive definite according to
P1 P A I ðu2 þ v 2 Þ;
ð55Þ
when (45) holds.
3. Stability criteria Theorem 1. Let (10)–(12), (46) and (48) or (46) and (50) or (46) and (52) hold together with the condition (iii) or (iv) of Lemma 2. Then the null solution of system (15) is nonlinearly, asymptotically, exponentially (locally) stable. Proof. By virtue of the hypotheses, there exist two positive constants h1 and h2 , such that
(
P1 6 h1 ðu2 þ v 2 Þ;
ð56Þ
P2 þ P3 6 h2 ðu2 þ v 2 Þ; and hence, from (22) and (56), one obtains that
dV h 6 ðu2 þ v 2 Þ þ jWj; ds 2
ð57Þ
with h ¼ infðh1 ; h2 Þð> 0Þ. The boundedness of w; ui (i ¼ 1; 2), implies that
jWj 6 Mðu þ v Þðuv Þ 6
pffiffiffi 3 M 2 2 ðu þ v 2 Þ2 ; 2
ð58Þ
with
M ¼ maxðjA3 þ A2 wj; jA1 þ A3 wjÞ:
ð59Þ
Hence, starting form (57), one obtains: 3 1 dV 6 d1 V þ d2 V 2 ¼ Vðd1 þ d2 V 2 Þ; ds
ð60Þ
with
d1 ¼
h ð> 0Þ; 2m1
d2 ¼
pffiffiffi M 2 3
2A2
ð> 0Þ:
ð61Þ
1
Then the assumption V 20 < dd12 implies, by recursive argument, that:
V 6 V 0 eds ;
1
d ¼ d1 d2 V 20 ð> 0Þ:
ð62Þ
Moreover, by virtue of (26), V and W ¼ ðu2 þ v 2 Þ are equivalent and in particular (119)–(121) are satisfied. Now, since all the hypotheses of the Liapunov (asymptotic) stability theorem are satisfied (cfr. Appendix A, (ii)), the null solution of (15) is nonlinearly, asymptotically, exponentially, locally stable. h Remark 7. By virtue of Lemmas 2–5 and Theorem 1, apart from the critical case IA 0, the conditions (46) or the equivalent conditions
A > 0;
ðAIÞ < 0;
appear to be the basic conditions to guarantee the stability of the null solution of (15).
ð63Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
6875
Theorem 2. Let (10)–(12) hold and let (46) hold by virtue of
ðu1 Þ P k1 ;
ðu2 Þ P k2 ;
ð64Þ
with ki ð¼ 1; 2Þ positive constants. Then
(
u1 P 1 ð1 u1 Þs¼0 e2ðk1 eÞs ; w þ u2 6 ðw þ u2 Þs¼0 e2ðk2 eÞs ;
ð65Þ
with 0 6 e < infðk1 ; k2 Þ, guarantee the (local) nonlinear asymptotic exponential stability of the null solution of (15). Proof. For the proof see [18]. h Remark 8. Obviously (15) cannot admit periodic solutions when the conditions guaranteeing the asymptotic stability of the null solution hold. Theorem 3. Let (10)–(12) and (46) hold and let us assume that
8 ðe1 u1 Þ > 0; ðe2 u2 Þ > 0; ðu1 þ u2 Þ > 0; > > > > > wþu wþu > e4ð1e1 ÞFðsÞ ; 8s > 0; < 1u12 < 1u12 s¼0 wþu2 u2 > > > wþ e4ð1e2 ÞGðsÞ ; 8s > 0; > 1u1 1u1 > s¼0 > > R Rs : s FðsÞ ¼ 0 u1 ðzÞ dz; GðsÞ ¼ 0 u2 ðzÞ dz; with
ð66Þ
ei ði ¼ 1; 2Þ constants such that ð1 e1 Þu1 þ ð1 e2 Þu2 > 0;
8s 2 Rþ ;
ð67Þ
then the zero solution of (15) is nonlinearly (locally) asymptotically exponentially stable. Proof. Requiring
(
2u1 l1 þ ddls1 < 2e1 u1 l1 ;
ð68Þ
2u2 l2 þ ddls2 < 2e2 u2 l2 ; one obtains that
(
l1 < l1 ð0Þe2ð1e1 ÞFðsÞ ; l2 < l2 ð0Þe2ð1e2 ÞGðsÞ :
ð69Þ
The hypotheses (66)4 and (66)5, are verified by
l1 ¼
1 w þ u2 2 ; 1 u1
l2 ¼
1 u1 w þ u2
12 :
ð70Þ
With this choice, the energy EðsÞ given by (19) has to satisfy
dE 1 < 2ðe1 u1 Þ l1 u2 þ 2ðe2 u2 Þ l2 v 2 þ U; ds 2
ð71Þ
with U given by (20). Setting
h2 ¼ 2 minððe1 u1 Þ ; ðe2 u2 Þ Þ;
ð72Þ
one obtains
dE < h2 E þ U: ds
ð73Þ
The boundedness of w; ui (i ¼ 1; 2), implies that
pffiffiffi 3 M0 2 2 jUj 6 M ðu þ v Þðuv Þ 6 ðu þ v 2 Þ2 ; 2
ð74Þ
M0 ¼ maxðjl1 j; jwl2 jÞ:
ð75Þ
0
with
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Hence, starting form (73) one obtains: 3 1 dE 6 d01 E þ d02 E2 ¼ Eðd01 þ d02 E2 Þ; ds
ð76Þ
with
d01 ¼
h2 ð> 0Þ; m3
pffiffiffi M0 2
d02 ¼ 1
3
ð> 0Þ:
ð77Þ
2m22
d0
Then the assumption E20 < d10 implies, by recursive argument, that: 2
0
E 6 E0 ed s ;
1
d0 ¼ d01 d02 E20 ð> 0Þ:
ð78Þ
Moreover, by virtue of (27), E and W ¼ ðu2 þ v 2 Þ are equivalent and in particular (119)–(121) are satisfied. Now, since all the hypotheses of the Liapunov (asymptotic) stability theorem are satisfied (cfr. Appendix A, (ii)), the null solution of (15) is nonlinearly, asymptotically, exponentially, locally stable. h Remark 9. We observe that (i) (66)3 is necessary for the consistence of (67), while (67) guarantees the consistence of (66)4 and (66)5; (ii) Theorem 3 does not require necessarily
ðu1 Þ > 0;
ðu2 Þ > 0;
ð79Þ
but can hold also if
u1 u2 < 0; 8s > 0; A > 0:
ð80Þ
In fact, let
(
u1 ¼ u2 ½uðsÞ þ 1; u2 < 0; u > 0; e1 ¼ 12 ; e2 ¼ 12 ; A > 0:
ð81Þ
Then
ð1 e1 Þu1 þ ð1 e2 Þu2 ¼
uþ1 2
3 2
u2 þ u2 > 0;
ð82Þ
is verified by
u > 2; 8s > 0:
ð83Þ
Theorem 4. Let (10)–(12) hold and suppose that (46) hold by virtue of
ðu1 Þ > 0;
ðu2 Þ > 0:
ð84Þ
On assuming that
ð1 u1 Þ ðw þ u2 Þ < ðu1 Þ ðu2 Þ ;
ð85Þ
the zero solution of (15) is nonlinearly (locally) asymptotically exponentially stable. Proof. Choosing
l1
1 ðw þ u2 Þ 2 ¼ ; ð1 u1 Þ
l2 ¼
ð1 u1 Þ ðw þ u2 Þ
12 ;
ð86Þ
it follows that
dE 6 u1 l1 u2 u2 l2 v 2 þ l1 ð1 u1 Þ þ l2 ðw þ u2 Þ uv þ U: ds
ð87Þ
In view of (85) and (86) one obtains
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dE 6 ðu1 Þ l1 u2 ðu2 Þ l2 v 2 þ 2 ð1 u1 Þ ðw þ u2 Þ uv þ U: ds
ð88Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
Since
6877
l1 ¼ l1 2 it turns out that 8 > < ð1 u1 Þ ðw þ u2 Þ < ðu1 Þ ðu2 Þ ¼ l1 ðu1 Þ l2 ðu2 Þ ; ð1 u1 Þ ðw þ u2 Þ ¼ g2 l1 ðu1 Þ l2 ðu2 Þ ; > : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 u1 Þ ðw þ u2 Þ uv 6 gðl1 ðu1 Þ u2 þ l2 ðu2 Þ v 2 Þ;
ð89Þ
with 0 < g ¼ const: < 1. Then (88) becomes
dE 6 ð1 gÞ ðu1 Þ l1 u2 þ l2 ðu2 Þ v 2 þ U; ds
ð90Þ
and hence
dE 6 h3 E þ U; ds
ð91Þ
h3 ¼ 2ð1 gÞ minððu1 Þ ; ðu2 Þ Þ:
ð92Þ
with
Following the same procedure used in Theorem 3, the thesis is hold. h Theorem 5. Let (10)–(12) hold together with (46) by virtue of
ðu1 Þ > 0;
ðu2 Þ > 0;
ð93Þ
and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 u1 Þ þ ðw þ u2 Þ < 2 ðu1 Þ ðu2 Þ :
ð94Þ
Then the null solution of system (15) is nonlinearly (locally) asymptotically exponentially stable. Proof. Choosing
l1 ¼ l2 ¼ 1, from (19) it follows that
dE 1 ¼ 2u1 u2 2u2 v 2 þ 2ðu1 1 þ u2 þ wÞuv þ U ds 2 6 ðu1 Þ u2 ðu2 Þ v 2 þ ðð1 u1 Þ þ ðw þ u2 Þ Þuv þ U:
ð95Þ
From (94), 9g ¼ const: 20; 1½ such that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 u1 Þ þ ðw þ u2 Þ ¼ 2g ðu1 Þ ðu2 Þ :
ð96Þ
Hence
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dE 6 ðu1 Þ u2 ðu2 Þ v 2 þ 2g ðu1 Þ ðu2 Þ uv þ U 6 h4 ð1 gÞðu2 þ v 2 Þ þ U; ds
ð97Þ
with h4 ¼ minððu1 Þ ; ðu2 Þ Þ. Then following the same procedure used in Theorem 3, the thesis is hold. h Theorem 6. Let (10)–(12) hold. Let us suppose that
A > 0;
I 0;
ð98Þ
together with (iii) or (iv) of Lemma 2. Then if
dA 6 0 8s 2 Rþ ; ds
ð99Þ
or if
dA ~ ¼ k; ds
~ ¼ const: > 0; k
then the null solution of (15) is nonlinearly (locally) asymptotically, exponentially, stable.
ð100Þ
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F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
Proof. From (17), by virtue of (98), one immediately obtains that V is positive definite. Moreover, from (22), on taking into account (ii) or (iv) of Lemma 2, it follows that
dV 6 hðu2 þ v 2 Þ þ jWj: ds Adopting the same procedure followed in Theorem 1, the thesis is hold.
h
Remark 10. When u1 ¼ u2 ¼ 0, (9) becomes
(
dX ds dY ds
¼ ð1 YÞX; ¼ wð1 þ XÞY;
that is a nonautonomous Lotka–Volterra model which has been analyzed by Rionero [21].
4. Instability criteria Instability criteria can be obtained, of course, by means either of the Liapunov function (17) or the function (16). We here recall – for the sake of completeness – the instability theorem obtained in [21] by the function (17) (cfr. Theorems 7–10) and concentrated ourselves on the instability theorems obtained by using the function (16). Theorem 7. Let (10)–(12), (41), (45) hold together with (i) of Lemma 2. Then the null solution of (15) is unstable. Theorem 8. Let (10)–(12), (53), (45) and (i) of Lemma 2 hold. Then the null solution of (15) is unstable. Theorem 9. Let
I 0;
ð101Þ
and (i) of Lemma 2 hold. Then if
dA ~ ¼ const > 0; Pk ds
ð102Þ
the null solution of (15) is unstable. Theorem 10. Let (10)–(12) hold. If
A 0;
ð103Þ
and (ii) of Lemma 2 hold, then the null solution of (15) is (Cetaev) unstable. Theorem 11. Let (41) hold by virtue of
u1 6 h1 ; u2 6 h2 ;
ð104Þ
with hi (i ¼ 1; 2) positive constants. Then
(
w þ u2 > ðw þ u2 Þs¼0 e2ðh1 eÞs ;
u1 < 1 ð1 u1 Þs¼0 e2ðh2 eÞs ;
ð105Þ
with 0 < e < infðh1 ; h2 Þ guarantee the instability of the null solution of (15). Proof. Choosing
l1 ¼ w þ u2 ; l2 ¼ 1 u1 ;
ð106Þ
it follows that
E¼
1 ðw þ u2 Þu2 þ ð1 u1 Þv 2 ; 2
ð107Þ
F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
6879
is positive definite and
dE 1 ¼ ds 2
dðw þ u2 Þ dð1 u1 Þ 2u1 ðw þ u2 Þ u2 þ 2u2 ð1 u1 Þ v 2 ; ds ds
ð108Þ
where we have disregarded the contribution of U. Then (12)2 and (105) guarantee that
( dðwþu
2Þ 2 ds dð1u1 Þ 2 ds
u1 ðw þ u2 Þ > 2eðw þ u2 Þ > 0; u2 ð1 u1 Þ > 2eð1 u1 Þ > 0:
ð109Þ
Hence all the hypotheses of the instability Liapunov theorem (cfr. Appendix A) are verified. h Theorem 12. Let (10)–(12) and (53) hold by virtue of
ju 2 j 6 h2 ;
ðu1 Þ P h1 ;
ð110Þ
with hi (i ¼ 1; 2) positive constants such that h1 > h2 . Then
(
w þ u2 > ðw þ u2 Þ0 e2ðeþh2 Þs ;
u1 > 1 þ ðu1 1Þ0 e2ðeh1 Þs ;
ð111Þ
imply the instability of the null solution of (15). Proof. Since (53) hold, we have to require
u1 u2 < ðu1 1Þðw þ u2 Þ þ A ; 8s 2 Rþ :
ð112Þ
In fact, if h1 > h2 it follows that
u1 þ u2 P h1 ju2 j P h1 h2 > 0; and hence I < 0. Choosing
l1 ¼ u1 1; l2 ¼ w þ u2 ;
ð113Þ
one has
E¼
1 ðu1 1Þu2 þ ðw þ u2 Þv 2 ; 2
ð114Þ
and
dE 1 ¼ ds 2
h i du1 dðw þ u2 Þ 2u1 ðu1 1Þ u2 þ 2u2 ðw þ u2 Þ v 2 þ 2 ðu1 1Þ2 þ ðw þ u2 Þ2 uv ; ds ds
ð115Þ
where we have disregarded the contribution of U. Hence
dE 1 P ds 2
dðu1 1Þ dðw þ u2 Þ 2u1 ðu1 1Þ u2 þ 2u2 ðw þ u2 Þ v 2 ; ds ds
ð116Þ
and the conditions (111) guarantee that
( dðu
u1 ðu1 1Þ > 2eðu1 1Þ > 0; u2 ðw þ u2 Þ > 2eðw þ u2 Þ > 0:
1 1Þ 2 ds dðwþu2 Þ 2 ds
ð117Þ
Hence E satisfies all the hypotheses of the Cetaev instability theorem (Appendix A). h
5. Final remarks (i) A class of generalized non-autonomous bidimensional Lotka–Volterra model of binary O.D.Es is introduced; (ii) The existence of biological meaningful equilibrium state is guaranteed together with the growing up of the preys in the absence of predators and predators decreasing in the absence of preys; (iii) The nonlinear stability of the equilibrium state is studied.
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F. Capone et al. / Applied Mathematics and Computation 219 (2013) 6868–6881
Acknowledgments This paper has been performed under the auspices of the G.N.F.M. of I.N.d.A.M. and Programma F.A.R.O. (Finanziamenti per l’ Avvio di Ricerche Originali, III tornata) ‘‘Controllo e stabilità di processi diffusivi nell’ambiente’’, Polo delle Scienze e Tecnologie, Università degli Studi di Napoli Federico II. Appendix A. Essential ingredients of the Liapunov Direct Method for non autonomous binary systems of O.D.Es We recall here the essential ingredients of the Liapunov Direct Method for nonautonomous binary systems of O.D.Es. (cfr [25, pp. 221–228]). Let Vðx; y; tÞ be a real single valued function – depending explicitly on t – defined in the space–time domain
Dðt0 ; rÞ : ðt; x; yÞ 2 D ft P t 0 ; x2 þ y2 6 rg;
ð118Þ
where t0 and r are constants such that ðt0 P 0; r > 0Þ. If – for a sufficiently large t 0 and a sufficiently small r – exists (an independent of t) function Wðx; yÞ, positive in Dðt0 ; rÞ, such that
Vð0; 0; tÞ ¼ 0;
8t P t 0 ;
Vðx; y; tÞ P Wðx; yÞ;
ð119Þ
ðV P Wðx; yÞÞ;
then V is said to be positive (negative) definite. V is said to be positive (negative) semidefinite if has only positive (negative) sign in D but can become zero at some point other than the origin. A function Vðx; y; tÞ said to admit an infinitely small upper limit if exists a t0 P 0 and r > 0 such that
jVj < Wðx; yÞ in Dðt 0 ; lÞ; 2
ð120Þ 2
with W definite in the disk x þ y 6 r and such that
lim Wðx; yÞ ¼ 0:
ð121Þ
x2 þy2 !0
The main stability theorems of the Direct Method for non autonomous systems can be summarized as follows Liapunov stability theorems: The existence of a positive function V implies (i) stability of the null solution if the time derivative of V along the solutions is negative semidefinite; (ii) asymptotic stability if V admits an upper bound which is infinitely small at the origin and V_ is negative definite. The main instability theorems for nonautonomous systems (cfr [25, pp. 226–227]) are Liapunov instability theorem: If exists a function V such that has an infinitely small upper limit and its time derivative V_ along the solutions is definite and also if for t P t 0 (with arbitrarily large t0 ) the function V can have the same sign as V_ in a neighborhood of the origin, then the null solution is unstable. Cetaev instability theorem: If for any disk D of the phase space centered at the origin, there exist a function V and an open subset H D, such that 8t P t 0 (i) (ii) (iii) (iv)
V is bounded in H and assumes positive values in H \ D, V vanishes on @ H \ D, the time derivative of V, along the solution, is positive definite in H, 0 2 @ H,
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