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On the dynamics of an n-dimensional ratio-dependent predator–prey system with diffusion
Applied Mathematics and Computation 208 (2009) 98–105
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On the dynamics of an n-dimensional ratio-dependent predator–prey system with diffusion Cosme Duque a, Krisztina Kiss b,1, Marcos Lizana c,* a b c
Departamento de Cálculo, Facultad de Ingenierı´a, Universidad de los Andes, Mérida, Venezuela Institute of Mathematics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary Departamento de Matemáticas, Facultad de Ciencias, Universidad de los Andes, Mérida, Venezuela
a r t i c l e
i n f o
Keywords: Dissipation persistence Ratio-dependent predator–prey system Reaction–diffusion system
a b s t r a c t The main concern of this paper is to study the dynamic of an n-dimensional ratio-dependent predator–prey system with diffusion. More concretely, we study the dissipativeness, the persistence of the system and we obtain condition under which the nontrivial equilibrium is globally asymptotically stable. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction There is a growing biological and physiological evidence [1–3,5] that in some situations, specially when predator have to search for food and therefore have to share or compete for food, a more suitable general predator prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. This is supported by numerous field and laboratory experiments and observations [2–4]. Hsu et al. in [8] performed a global analysis of the Michaelis–Menten-type ratio-dependent predator–prey system without diffusion. Concretely, they study the following system:
sNP N0 ðtÞ ¼ Nð1 NÞ ; PþN N : P0 ðtÞ ¼ dP r þ NþP Moreover, they discuss the main differences between the classical predator–prey and the ratio-dependent predator–prey models. In particular they brought into discussion the well-known ‘‘paradox of enrichment” or equivalently ‘‘the biological control paradox”. Kiss and Kovács generalized and studied this model for n predators and one prey species, see [9]. Now, if the predator and the prey are confined to a fixed bounded domain X in RN with smooth boundary, and their densities are spatially inhomogeneous, it is much suitable to consider a reaction diffusion system, subject to homogeneous Neumann boundary conditions. In this paper, we will consider the following n-dimensional ratio-dependent predator–prey system n @u u X mi uv i ¼ d0 Du þ ru 1 ; @t K a iv i þ u i¼1
@v i mi uv i ¼ di Dv i ai v i þ ; @t ai v i þ u
i ¼ 1; . . . ; n;
* Corresponding author. E-mail addresses: [email protected] (C. Duque), [email protected] (K. Kiss), [email protected] (M. Lizana). 1 The present work has partially been supported by the Hungarian National Scientific Research Foundation No. NK 63066. 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.016
ð1Þ
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
99
where x 2 X; t > 0. The spatial population densities of the prey and the predator species are respectively denoted by uðx; tÞ and v i ðx; tÞ; D denotes the Laplacian operator, x 2 X; t > 0; di ; r; K; mi ; ai ; ai are positive constants. In order to study the solution of (1) one has to specify initial conditions of the form
v i ðx; 0Þ ¼ ui ðxÞ;
uðx; 0Þ ¼ u0 ðxÞ;
i ¼ 1; . . . ; n; x 2 X;
ð2Þ
where ui ; i ¼ 0; . . . ; n are non-negative continuous functions and homogeneous Neumann boundary conditions which we assume to be
@u @ v i ¼ ¼ 0; @g @g
x 2 @ X; t P 0; i ¼ 1; . . . ; n;
ð3Þ
where @@g denotes the outward normal derivative on @ X; and X RN is a bounded domain with smooth boundary. For simplicity, we nondimensionalize the system (1) with the scaling t ! rt; u ! u=K; v i ! ai v i =K; i ¼ 1; . . . ; n. Then system (1) takes the form n X @u uv i si ¼ D0 Du þ uð1 uÞ ; @t v i þu i¼1
@v i uv i ¼ Di Dv i di v i þ bi ; @t vi þ u
ð4Þ
i ¼ 1; . . . ; n;
where
D0 ¼
d0 ; r
Di ¼
di ; r
si ¼
mi rai
di ¼
ai r
bi ¼
mi ; r
i ¼ 1; . . . ; n:
The main concern of this paper is to study the dynamic of the system (4). More concretely, in the following section we will show that the system (4) generates a semi dynamical system in a suitable Banach space and it is point dissipative. Furthermore, we study the persistence of the system (4). Finally, we prove the global stability of the unique nontrivial equilibrium of the system (4), for certain parameter’s configuration. 2. Preliminaries: dissipativeness of the reaction diffusion system We will show that the reaction–diffusion system (4) generates a semi dynamical system and it is biologically well possed on suitable Banach space. Let us set F ¼ ðf0 ; f1 ; . . . ; fn Þ; U ¼ ðu; v 1 ; . . . ; v n Þ and D ¼ diag½D0 ; D1 ; . . . ; Dn , where
f0 ¼ uð1 uÞ
n X i¼1
fi ¼ di v i þ bi
si
uv i ; vi þ u
uv i ; vi þ u
ð5Þ
i ¼ 1; . . . ; n:
ð6Þ
Henceforth, considering also an initial condition, system (4) can be rewritten as
@Uðx; tÞ ¼ DDUðx; tÞ þ FðUÞ; x 2 X; @t @U ðx; tÞ ¼ 0; x 2 @ X; t > 0 @g Uðx; 0Þ ¼ uðxÞ ¼ ðu0 ; . . . ; un Þ;
t>0 ð7Þ
x 2 X:
Let X be the Banach space X 0 X n , where X i ¼ CðXÞ; i ¼ 0; . . . ; n. The norm on X is defined by juj ¼ ju0 j þ þ jun j. Let A0i ; i ¼ 0; . . . ; n be the differential operators A0i wi ¼ Di Dwi , defined on the domains DðA0i Þ; i ¼ 0; . . . ; n; where
DðA0i Þ ¼
@wi wi 2 C 2 ðXÞ \ C 1 ðXÞ : A0i wi 2 CðXÞ; ðxÞ ¼ 0; x 2 @ X ; @g
i ¼ 0; . . . ; n:
The closures Ai of A0i in X i generate analytic semigroups of bounded linear operators T i ðtÞ; i ¼ 0; . . . ; n for t P 0 such that wi ðtÞ ¼ T i ðtÞui are solutions of the abstract linear differential equations in X i given by
w0i ðtÞ ¼ Ai wi ðtÞ;
i ¼ 0; . . . ; n:
An additional property of the semigroup is that for each t > 0; T i ðtÞ; i ¼ 0; . . . ; n are compact operators. Let TðtÞ : X ! X be defined by TðtÞ ¼ T 0 ðtÞ T n ðtÞ. Then TðtÞ is a semigroup of operators on X generated by the operator A ¼ A0 An defined on DðAÞ ¼ DðA0 Þ DðAn Þ and Uðx; tÞ ¼ ½TðtÞuðxÞ is the solution of the linear system
@U ðx; tÞ ¼ DDUðx; tÞ; x 2 X; t > 0 @t @U ðx; tÞ ¼ 0; x 2 @ X; t > 0; Uðx; 0Þ ¼ uðxÞ; x 2 X: @g
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C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
In the language of partial differential equations Uðx; tÞ ¼ ½TðtÞuðxÞ are classical solutions of the initial boundary value problem (5), with F ¼ 0. Observe that the nonlinear term F is twice continuously differentiable. Therefore, we can define the map ½F ðuÞðxÞ ¼ FðuðxÞÞ which maps X into itself and equation (5) can be viewed as the abstract ODE in X given by
w0 ðtÞ ¼ AwðtÞ þ F ðwðtÞÞ;
wð0Þ ¼ u;
w ¼ ðw0 ; . . . ; wn Þ:
ð8Þ
While a solution wðtÞ of (6) can be obtained under the restriction that u 2 DðAÞ, a mild solution can be obtained for every u 2 X by requiring only that wðtÞ is a continuous solution of the following integral equation:
wðtÞ ¼ TðtÞu þ
Z
t
Tðt sÞF ðwðsÞÞds;
t 2 ½0; bÞ;
0
where b ¼ bðuÞ 6 1. Restricting our attention to functions u in the set
X K ¼ fu 2 X : uðxÞ 2 K; x 2 Xg; nþ1 where K ¼ Rþ , and taking into account the definition of the functions fi , we obtain that
f0 ð0; v 1 ; . . . ; v n Þ ¼ 0;
f i ðu; v 1 ; . . . ; v i1 ; 0; v iþ1 ; . . . ; v n Þ ¼ 0; i ¼ 1; . . . ; n
for U 2 K. Thus, Corollary 3.2, p. 129 in [10] implies that the Nagumo condition for the positive invariance of K is satisfied, i.e. 1
lim h distðK; U þ hFðUÞÞ ¼ 0;
U 2 K:
h!0þ
ð9Þ
On the other hand, a direct applications of the strong parabolic maximum principle can be used to show that the linear semigroup TðtÞ leaves X K positively invariant, i.e.
TðtÞX K X K ;
t P 0:
ð10Þ
Remark 1. Let us pointing out that the invariance of the set X K can be obtained directly, just having in mind the special structure of the vector field F. Indeed, the coordinate hyperplanes u ¼ 0; v i ¼ 0; i ¼ 1; . . . ; n are invariant under the semiflow induced by the system (1). Now, we are going to prove that all solutions of the system (4) are bounded and therefore defined for all t P 0. Theorem 2.1. (Dissipativeness). Let ðu; v 1 ; . . . ; v n Þ be any solution of (4). Then
lim sup max uðx; tÞ 6 1; t!1
lim sup max v i ðx; tÞ 6 maxf0; t!1
ð11Þ
x2X
x2X
ðbi di Þ g; di
i ¼ 1; . . . ; n:
ð12Þ
Proof. From the first equation of the system (4), it follows that
@u D0 Du 6 uð1 uÞ; @t as long as u is defined as a function of t. Let z be the solution of the equation
z0 ðtÞ ¼ zð1 zÞ; zð0Þ ¼ max uðx; 0Þ: x2X
From the comparison principle (see Theorem 10.1, p. 94 in [11]), we obtain uðx; tÞ 6 zðtÞ. Now, taking into account that for any > 0 there exists a T > 0 such that zðtÞ < 1 þ for any t P T , which in turn implies that uðx; tÞ is defined for all t P 0, and
lim sup max uðx; tÞ 6 1: t!1
x2X
Having in mind that for a given > 0 there exists a T > 0 such that uðx; tÞ 6 1 þ for any x 2 X and t P T , and by using the second equation of (4), we get
@v i u 1þ ð1 þ Þðbi di Þ di v i 6 v i di þ bi ¼ vi Di Dv i ¼ v i di þ bi u þ vi @t 1 þ þ vi 1 þ þ vi for any x 2 X and t P T . If bi > di , let zi be the solution of the following initial value problem
z0i ðtÞ ¼
zi ½ð1 þ Þðbi di Þ di zi ; 1 þ þ zi
zi ðT Þ ¼ max v i ðx; T Þ > 0: x2X
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
101
Then
lim zi ðtÞ ¼
t!þ1
ð1 þ Þðbi di Þ : di
Hence, by using the comparison principle, we obtain that
v i ðx; tÞ 6 zi ðtÞ; which implies that
ðb di Þ lim sup max v i ðx; tÞ 6 i : di t!1 x2X If bi 6 di , we have
di v 2i @v i ; Di Dv i 6 @t 1 þ þ vi which implies that
lim sup max v i ðx; tÞ 6 0; t!1
i ¼ 1; . . . ; n:
x2X
This completes the proof of our claim.
h
Remark 2. From the proof of the dissipativeness, we obtain as a corollary that the predators cannot survive provided that bi 6 di , for i ¼ 1; . . . ; n. In the original variables, this means that the necessary condition for the survival of each predator the birth rate mi have to be larger than the death rate ai , i.e. mi > ai . Remark 3. Another immediate consequence of the proof of the former result is that for a given fu 2 X K : uðxÞ 2 Bg is an absorbing set for the system (4), where
> 0; A ¼
ðb d1 Þ ðb dn Þ B ¼ ½0; 1 þ 0; 1 þ 0; n þ : d1 dn
Finally, Theorem 2.1, conditions (8) and (9) together allow us apply Theorem 3.1, p. 127 in [10], giving us Lemma 2.1. For each u 2 X K , (4) has a unique mild solution wðtÞ ¼ wðu; tÞ 2 X K and a classical solution Uðx; tÞ ¼ ½wðtÞðxÞ. Moreover, the set X K is positively invariant under flow Wt ðuÞ ¼ wðu; tÞ induced by (4). So, the model (4) is biologically well possed. Moreover, from the previous theorem and by using the Theorem 3.4.8, p. 40, in [7], it follows that the relevant dynamic of the system (4) is concentrated on a compact set of the space X K , which is contained in A . 3. Uniform persistence of the predators The goal of this section is to give conditions implying that the predators and prey persist indefinitely, i.e., that neither becomes extinct. Definition 3.1. Problem (4) is said to have the persistence property if, for any non-negative initial data uðxÞ, with ui ðxÞX0; i ¼ 0; . . . ; n; there exists a positive constant e ¼ eðu) such that the corresponding solution ðu; v 1 ; . . . ; v n Þ of (1) satisfies
lim inf min uðx; tÞ P e; t!1 x2X
lim inf min v i ðx; tÞ P e; i ¼ 1; . . . ; n: t!1 x2X
Theorem 3.1. If bi > di ; i ¼ 1; . . . ; n and
Pn
i¼1 si
< 1, then (4) has the persistence property.
Proof. Let us suppose that ui ðxÞ > 0; i ¼ 0; . . . ; n. From the first equation of the system (4), we obtain n n n X X X @u si uv i si þ u2 D0 Du ¼ uð1 uÞ ¼ uð1 uÞ u @t vi þ u i¼1 i¼1 i¼1
Since
Pn
i¼1 si
< 1, it follows by a comparison argument that
lim inf min uðx; tÞ P 1 t!1
X
n X
si > 0:
i¼1
Thus, there exists a T > 0 such that
uðx; tÞ P
" # n X 1 si ¼ g > 0 8x 2 X; t P T: 1 2 i¼1
! n X si si : Pu 1u vi þ u i¼1
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C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
Taking into account this inequality and by using the second equation of (4), we get
@v i u g gðbi di Þ di v i Di Dv i ¼ v i di þ bi P v i di þ bi P vi g þ vi u þ vi @t g þ vi for any x 2 X and t P T. Let zi be the solution of the following initial value problem:
z0i ðtÞ ¼
zi ½gðbi di Þ di zi ; g þ zi
zi ðTÞ ¼ min v i ðx; T Þ > 0: x2X
Then
lim zi ðtÞ ¼
gðbi di Þ di
t!þ1
> 0;
since bi > di . Hence, by using the comparison principle, we obtain that
lim inf min v i ðx; tÞ P
gðbi di Þ
t!1 x2X
di
;
v i ðx; tÞ P zi ðtÞ; which implies that
i ¼ 1; . . . ; n:
Which in turn completes the proof of our assertion.
h
Let us finish this section with a couple of remarks. Remark 4. In the theorem 3.1, it might be possible to relax the assumption u1 ðxÞ > 0 for all x 2 X (which is required so that zð0Þ > 0), to u1 ðxÞ P 0 for all x 2 X with u1 ðxÞX0. In this later situation the strong maximum principle implies u1 ðx; tÞ > 0 for all t > 0; x 2 X. One could change the origin of time to a positive time t1 and still get zð0Þ > 0 for an appropriately modified comparison problem. Remark 5. It is interesting to point out, that under the conditions for the persistence of the system (4), the interaction matrix of system (4) is sign stable, namely the coefficient matrix of the ODE connection with (4) linearized at ðu ; v 1 ; v 2 ; . . . ; v n Þ is sign stable. For more details we refer the reader to Theorem 2.2. in [9]. 4. Global stability of the nontrivial equilibrium The main concern of this section is to prove the global stability of ðu ; v 1 ; . . . ; v n Þ, the unique nontrivial equilibrium of the system (4), where the coordinates of the equilibrium are given by
u ¼ 1
n X si ðbi di Þ ; bi i¼1
v i ¼
ðbi di Þ u ; i ¼ 1; . . . ; n: di
In order to achieve our goal, we will use basically the method of upper and lower solutions combined with the monotone iteration method. More concretely, we are going to construct 2n þ 2 sequences, namely ð1Þ ð2Þ ð1Þ ð2Þ fxk g; fxk g; fyi;k g; fyi;n g for i ¼ 1; . . . ; n; which satisfy the following properties: ð1Þ
ð2Þ
(i) xk 6 lim inf t!1 minX uðx; tÞ 6 lim supt!1 maxX uðx; tÞ 6 xk , (ii) (iii) (iv)
ð1Þ yi;k 6 lim inf t!1 minX i ðx; tÞ 6 lim supt!1 maxX i ðx; tÞ ð1Þ ð1Þ xk ; yi;k ; i ¼ 1; . . . ; n, are non-decreasing sequences. ð2Þ ð2Þ xk ; yi;k i ¼ 1; . . . ; n, are non-increasing sequences.
v
v
ð2Þ
6 yi;k ; i ¼ 1; . . . ; n,
Let us set
x1 ¼ 1
n X
si ;
x2 ¼ 1;
i¼1
yi;1 ¼ g
bi di ; di
yi;2 ¼
bi di ; di
i ¼ 1; . . . ; n:
ð13Þ
From theorems 2.1 and 3.1 we have that the following bounded closed domain
H ¼ fðu; v 1 ; . . . ; v n Þ : ðu; v 1 ; . . . ; v n Þ 2 ½x1 ; x2 ½y1;1 ; y1;2 ½yn;1 ; yn;2 g is eventually invariant for system (4), i.e., for any positive solution ðu; v 1 ; . . . ; v n Þ of the system (4), the following inequalities hold:
1
n X
si 6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 1;
i¼1
g
t!1
X
t!1
bi di b di 6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 i t!1 di di t!1 X X
for i ¼ 1; . . . ; n.
ð14Þ
X
ð15Þ
103
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
Let us start constructing the sequences, recurrently. Let us set ð1Þ
ð2Þ
x1 ¼ x 1 ;
ð1Þ
x1 ¼ x2 ;
ð2Þ
yi;1 ¼ yi;1 ;
yi;1 ¼ yi;2 ;
i ¼ 1; . . . ; n;
where x1 ; x2 ; yi;1 ; yi;2 ; i ¼ 1; . . . ; n, were defined in (13). From (14) and (15), it follows that for any exists a T > 0 such that ð1Þ
e > 0 small enough, there
ð2Þ
x1 e 6 uðx; tÞ 6 x1 þ e ð1Þ
ð2Þ
yi;1 e 6 v i ðx; tÞ 6 yi;1 þ e;
ð16Þ
i ¼ 1; . . . ; n
for any t > T, uniformly for x 2 X. First of all, let us observe that (16) implies that
u
616
ð2Þ
x1 þ e
vi
vi
and
ð1Þ
yi;1 e
ð2Þ
yi;1 þ e
616
u
:
ð1Þ
x1 e
From these inequalities, it is not difficult to obtain the following estimations ð1Þ
yi;1 e ð1Þ
ð2Þ
6
yi;1 þ x1
ð2Þ
vi u þ vi
yi;1 þ e
6
ð2Þ
ð1Þ
yi;1 þ x1
ð17Þ
:
By using the first equation of the system (4), and the estimations (16) and (17), we obtain that
" # ð1Þ n n X X yi;1 e @u uv i si 6u 1 si ð1Þ u ; D0 Du ¼ uð1 uÞ ð2Þ @t u þ vi y þ x1 i¼1 i¼1
t > T; x 2 X;
" # ð2Þ n n X X yi;1 þ e @u uv i si Pu 1 si ð2Þ u ; D0 Du ¼ uð1 uÞ ð1Þ @t u þ vi y þ x1 i¼1 i¼1
t > T; x 2 X:
i;1
and
i;1
Applying comparison techniques to each of the above problems, we certainly obtain that
1
n X
ð2Þ
si
i¼1
yi;1 þ e ð2Þ
6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 1
ð1Þ
t!1
yi;1 þ x1
t!1
X
X
n X
ð1Þ
si
i¼1
yi;1 e ð1Þ
ð2Þ
yi;1 þ x1
:
Now, from the second equation of the system (4), estimation (16) and the comparison principle, we get
ðbi di Þ ð1Þ ðb di Þ ð2Þ ðx1 eÞ 6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 i ðx1 þ eÞ t!1 di di t!1 X X for i ¼ 1; . . . ; n. Since e is arbitrary positive, we may assert that the following inequalities hold:
1
n X
ð2Þ
si
i¼1
yi;1 ð2Þ
6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 1
ð1Þ
t!1
yi;1 þ x1
t!1
X
X
n X i¼1
ð1Þ
si
yi;1 ð1Þ
ð2Þ
yi;1 þ x1
;
ðbi di Þ ð1Þ ðb di Þ ð2Þ x1 6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 i x1 t!1 di di t!1 X X for i ¼ 1; . . . ; n. Now, we are in position to define the second terms of the sequences under construction, setting ð1Þ
x2 ¼ 1
n X
ð2Þ
si
i¼1 ð1Þ
yi;2 ¼
yi;1 ð2Þ yi;1
ðbi di Þ ð1Þ x1 ; di
þ
ð2Þ
ð1Þ x1
; x2 ¼ 1
n X i¼1
ð2Þ
yi;2 ¼
ð1Þ
si
yi;1 ð1Þ yi;1
ðbi di Þ ð2Þ x1 ; di
ð2Þ
þ x1
;
i ¼ 1; . . . ; n:
Repeating the above performed process, we obtain the recurrence to construct the sequences n o ð2Þ yi;k ; i ¼ 1; . . . ; n. Actually, they are defined as follows: ð1Þ
x1 ¼ x 1 ;
ð1Þ
n X
ð2Þ
n X
xk ¼ 1
si
i¼1 ð2Þ
x1 ¼ x 2 ;
xk ¼ 1
i¼1 ð1Þ
yi;1 ¼ yi;1 ; for i ¼ 1; . . . ; n.
ð1Þ
yi;k ¼
ð2Þ yi;k1 ð2Þ ð1Þ yi;k1 þ xk1 ð1Þ
si
n o n o n o ð1Þ ð2Þ ð1Þ xk ; xk ; yi;k , and
yi;k1 ð1Þ
ð2Þ
yi;k1 þ xk1
ðbi di Þ ð1Þ xk ; di
ð2Þ
ð18Þ
;
yi;1 ¼ yi;2 ;
ð2Þ
yi;k ¼
ðbi di Þ ð2Þ xk di
104
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
Remark 6. It is worthwhile to point out that from the constructing method of the sequences, they satisfy automatically the properties (i) and (ii) enumerated above. ð1Þ
ð1Þ
ð2Þ
ð2Þ
Proposition 4.1. The sequences xk ; yi;k are non-decreasing sequences; and, xk ; yi;k are non-increasing sequences. ð1Þ
ð1Þ
ð2Þ
ð2Þ
Proof. The proof we will carry out by induction. From the definition of the first terms of the sequences xk ; yi;k ; xn ; yi;k , without any additional restriction on the parameters, we immediately get ð1Þ
x1 ¼ 1
n X
si 6 1
n X
i¼1 ð2Þ
x2 ¼ 1
ð2Þ
yi;2
yi;1
ð1Þ
ð2Þ
ð1Þ
i¼1
yi;1 þ x1
ð2Þ
6 1 ¼ x1 ;
¼ x2 ;
ð1Þ
n X
si
i¼1 ð1Þ yi;1
ð2Þ
si
yi;1
ð2Þ
ð1Þ
yi;1 þ x1
ðb di Þ ðbi di Þ ð1Þ ð1Þ ¼g i 6 x1 ¼ yi;2 di di ðb di Þ ð2Þ ðbi di Þ ð2Þ ¼ i x1 6 ¼ yi;1 di di
for i ¼ 1; . . . ; n. Let us assume now that our claim is true for m; i.e. ð1Þ
ð2Þ
xm1 6 xð1Þ m ;
ð1Þ
xð2Þ m 6 xm1 ;
ð1Þ
ð2Þ
yi;m1 6 yi;m ;
ð2Þ
yi;m 6 yi;m1 :
After a tedious but straightforward computation, we obtain that
xð1Þ m ¼ 1
n X
ð2Þ
si
i¼1 ð2Þ
xmþ1 ¼ 1
n X
yi;m1 ð2Þ
ð1Þ yi;m
ð1Þ
ð2Þ
yi;mþ1
þ
ð2Þ xm
n X
ð2Þ
si
i¼1
ð1Þ
yi;m
si
i¼1
yi;m ¼
ð1Þ
yi;m1 þ xm1
61
61
n X
yi;m
ð1Þ
ð2Þ
¼ xmþ1 ;
ð1Þ
yi;m þ xm
ð1Þ yi;m1 ð1Þ ð2Þ yi;m1 þ xm1
si
i¼1
¼ xð2Þ m ;
ðbi di Þ ð1Þ ðb di Þ ð1Þ ð1Þ xm1 6 i xm ¼ yi;mþ1 ; di di ðb di Þ ð2Þ ðbi di Þ ð2Þ ð2Þ ¼ i xm 6 xm1 ¼ yi;m di di
for i ¼ 1; . . . ; n. This complete the proof of our assertion.
h
Proposition 4.1 implies that all sequences in (18) are convergent. Let us denote their limits by ð1Þ
ð2Þ
lim xk ¼ xð1Þ ;
lim xk ¼ xð2Þ ;
k!1
k!1
ð1Þ
ð1Þ
ð2Þ
lim yi;k ¼ yi ;
ð2Þ
lim yi;k ¼ yi
k!1
k!1
for i ¼ 1; . . . ; n. So, taking into account the definition of the sequences and letting k ! 1, we obtain that
xð1Þ ¼ 1
n X i¼1
ð1Þ
yi
¼
n X
ð2Þ
si
yi ð2Þ
yi þ xð1Þ
ðbi di Þ ð1Þ x ; di
ð2Þ
yi
;
xð2Þ ¼ 1
ð1Þ
si
i¼1
¼
yi ð1Þ
yi þ xð2Þ
; ð19Þ
ðbi di Þ ð2Þ x di
for i ¼ 1; . . . ; n. Which immediately imply that
x1 6 xð1Þ 6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 xð2Þ 6 x2 ; t!1
yi;1 6
ð1Þ yi
X
t!1
X ð2Þ
6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 yi t!1
X
t!1
X
6 yi;2 ;
i ¼ 1; . . . ; n. Theorem 4.1. If
Pn
i¼1 si
< 1=3, then the unique positive constant equilibrium ðu ; v 1 ; . . . ; v n Þ of system (4) is a global attractor. ð1Þ
Proof. In order to prove this theorem, it is enough to show that xð1Þ ¼ xð2Þ ¼ u , and yi (19), we have,
ð2Þ
¼ yi
¼ v i ; i ¼ 1; . . . ; n. Indeed, from
105
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
xð1Þ ¼ 1
n X i¼1
x
ð2Þ
¼1
n X i¼1
si ½ðbi di Þ=di xð2Þ ; ½ðbi di Þ=di xð2Þ þ xð1Þ
ð20Þ
si ½ðbi di Þ=di xð1Þ : ½ðbi di Þ=di xð1Þ þ xð2Þ
Let us ni ¼ ðbi di Þ=di , now from (20),
xð1Þ xð2Þ ¼ ðxð1Þ xð2Þ Þ
n X i¼1
ðni
si ni ðxð1Þ þ xð2Þ Þ : þ xð2Þ Þðni xð2Þ þ xð1Þ Þ
xð1Þ
If xð1Þ < xð2Þ , then
1¼
n X i¼1
n n n n X X X X si ni ðxð1Þ þ xð2Þ Þ si ni si ni 2si ni < 6 þ < ð1Þ þ xð2Þ Þ ð2Þ þ xð1Þ Þ ðni xð1Þ þ xð2Þ Þðni xð2Þ þ xð1Þ Þ ðn x ðn x ðn þ 1Þx 1 i i i i¼1 i¼1 i¼1 i¼1
2si < 1; n P 1 si i¼1
P ð1Þ ð2Þ since ni¼1 si < 1=3. Contradiction. So xð1Þ ¼ xð2Þ and together with (19) yi ¼ yi , i ¼ 1; . . . ; n. ð1Þ ð1Þ Finally, by using (19), we obtain that x ¼ u and yi ¼ v i ; i ¼ 1; . . . ; n. This complete the proof. h 5. Discussion and remarks Let us start pointing out that for one predator prey systems with delay but without diffusion Tang et al. in [12] and for one predator prey systems with diffusion but without delay Fan et al. in [6] discussed the dynamic of the model and in particular the global stability of the nontrivial equilibrium. In both cases they considered a functional response of a ratio-dependent type. Later on, Kiss and Kovács in [9] studied the main features of a model of n predators and one prey species without diffusion, there they performed a thorough analysis of the stability of the nontrivial equilibrium. In this paper we considered a model of n predators and one prey species with diffusion and Michaelis–Menten-type ratio-dependent functional response. We were able to identify the region of persistence of the model and obtain conditions that guaranty the global stability of the nontrivial equilibrium of the model. Having in mind [9], it is easy Pn to show that if i¼1 si < 1=3; i.e., when the nontrivial equilibrium of the system (4) is a global attractor, then the interaction matrix of system (4) without diffusion is sign stable, namely the coefficient matrix of the ODE connection with (4) linearized at ðu ; v 1 ; v 2 ; . . . ; v n Þ is sign stable. This is quite natural from the biological and mathematical point of view. A lot questions remain without answer. Actually, the connection between the stability of the unique nontrivial equilibrium for the model with and without diffusion. In other words, patterns may arise for this model. Another very interesting question is the study of the impact of the dynamic that emerges from the trivial equilibrium. Let us remind that the vector field can be extended by continuity to the origin. This allows us to consider the origin as an equilibrium point of the model. All these question will be analyzed in a forthcoming paper. References [1] H.R. Akcakaya, R. Arditi, L.R. Ginzburg, Ratio-dependent prediction: an abstraction that works, Ecology 76 (1995) 995–1004. [2] R. Arditi, A.A. Berryman, The biological paradox, Trends Ecol. Evol. 6 (1991) 32. [3] R. Arditi, L.R. Ginzburg, H.R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent models, Am. Nat. 138 (1991) 1287– 1296. [4] R. Arditi, L.R. Ginzburg, Coupling in predator–prey dynamics: ratio-dependence, J. Theor. Biol. 139 (1989) 311–326. [5] C. Cosner, D.L. DeAngelis, J.S. Ault, D.B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol. 56 (1999) 65–75. [6] Y.H. Fan, W.T. Li, Global asymptotic stability of a ratio-dependent predator–prey system with diffusion, J. Comput. Appl. Math. 188 (2006) 205–227. [7] J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 1988. Number 25. [8] S.B. Hsu, T.W. Hwang, Y. Kuang, Global analysis of the Michaelis–Menten-type ratio-dependent predator–prey system, J. Math. Biol. 42 (2001) 489– 506. [9] K. Kiss, S. Kovács, Qualitative behaviour of n-dimensional ratio-dependent predator–prey systems, Appl. Math. Comput. 199 (2) (2008) 535–546. [10] L. Smith Hal, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, 1995. [11] J. Smoller, Shock Waves and Reaction–Diffusion Equations, second ed., Springer-Verlag, New York, 1994. [12] S. Tang, L. Chen, Global qualitative analysis for a ratio-dependent predator–prey model with delay, J. Math. Anal. Appl. 266 (2002) 401–419.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On the dynamics of an n-dimensional ratio-dependent predator–prey system with diffusion Cosme Duque a, Krisztina Kiss b,1, Marcos Lizana c,* a b c
Departamento de Cálculo, Facultad de Ingenierı´a, Universidad de los Andes, Mérida, Venezuela Institute of Mathematics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary Departamento de Matemáticas, Facultad de Ciencias, Universidad de los Andes, Mérida, Venezuela
a r t i c l e
i n f o
Keywords: Dissipation persistence Ratio-dependent predator–prey system Reaction–diffusion system
a b s t r a c t The main concern of this paper is to study the dynamic of an n-dimensional ratio-dependent predator–prey system with diffusion. More concretely, we study the dissipativeness, the persistence of the system and we obtain condition under which the nontrivial equilibrium is globally asymptotically stable. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction There is a growing biological and physiological evidence [1–3,5] that in some situations, specially when predator have to search for food and therefore have to share or compete for food, a more suitable general predator prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. This is supported by numerous field and laboratory experiments and observations [2–4]. Hsu et al. in [8] performed a global analysis of the Michaelis–Menten-type ratio-dependent predator–prey system without diffusion. Concretely, they study the following system:
sNP N0 ðtÞ ¼ Nð1 NÞ ; PþN N : P0 ðtÞ ¼ dP r þ NþP Moreover, they discuss the main differences between the classical predator–prey and the ratio-dependent predator–prey models. In particular they brought into discussion the well-known ‘‘paradox of enrichment” or equivalently ‘‘the biological control paradox”. Kiss and Kovács generalized and studied this model for n predators and one prey species, see [9]. Now, if the predator and the prey are confined to a fixed bounded domain X in RN with smooth boundary, and their densities are spatially inhomogeneous, it is much suitable to consider a reaction diffusion system, subject to homogeneous Neumann boundary conditions. In this paper, we will consider the following n-dimensional ratio-dependent predator–prey system n @u u X mi uv i ¼ d0 Du þ ru 1 ; @t K a iv i þ u i¼1
@v i mi uv i ¼ di Dv i ai v i þ ; @t ai v i þ u
i ¼ 1; . . . ; n;
* Corresponding author. E-mail addresses: [email protected] (C. Duque), [email protected] (K. Kiss), [email protected] (M. Lizana). 1 The present work has partially been supported by the Hungarian National Scientific Research Foundation No. NK 63066. 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.016
ð1Þ
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
99
where x 2 X; t > 0. The spatial population densities of the prey and the predator species are respectively denoted by uðx; tÞ and v i ðx; tÞ; D denotes the Laplacian operator, x 2 X; t > 0; di ; r; K; mi ; ai ; ai are positive constants. In order to study the solution of (1) one has to specify initial conditions of the form
v i ðx; 0Þ ¼ ui ðxÞ;
uðx; 0Þ ¼ u0 ðxÞ;
i ¼ 1; . . . ; n; x 2 X;
ð2Þ
where ui ; i ¼ 0; . . . ; n are non-negative continuous functions and homogeneous Neumann boundary conditions which we assume to be
@u @ v i ¼ ¼ 0; @g @g
x 2 @ X; t P 0; i ¼ 1; . . . ; n;
ð3Þ
where @@g denotes the outward normal derivative on @ X; and X RN is a bounded domain with smooth boundary. For simplicity, we nondimensionalize the system (1) with the scaling t ! rt; u ! u=K; v i ! ai v i =K; i ¼ 1; . . . ; n. Then system (1) takes the form n X @u uv i si ¼ D0 Du þ uð1 uÞ ; @t v i þu i¼1
@v i uv i ¼ Di Dv i di v i þ bi ; @t vi þ u
ð4Þ
i ¼ 1; . . . ; n;
where
D0 ¼
d0 ; r
Di ¼
di ; r
si ¼
mi rai
di ¼
ai r
bi ¼
mi ; r
i ¼ 1; . . . ; n:
The main concern of this paper is to study the dynamic of the system (4). More concretely, in the following section we will show that the system (4) generates a semi dynamical system in a suitable Banach space and it is point dissipative. Furthermore, we study the persistence of the system (4). Finally, we prove the global stability of the unique nontrivial equilibrium of the system (4), for certain parameter’s configuration. 2. Preliminaries: dissipativeness of the reaction diffusion system We will show that the reaction–diffusion system (4) generates a semi dynamical system and it is biologically well possed on suitable Banach space. Let us set F ¼ ðf0 ; f1 ; . . . ; fn Þ; U ¼ ðu; v 1 ; . . . ; v n Þ and D ¼ diag½D0 ; D1 ; . . . ; Dn , where
f0 ¼ uð1 uÞ
n X i¼1
fi ¼ di v i þ bi
si
uv i ; vi þ u
uv i ; vi þ u
ð5Þ
i ¼ 1; . . . ; n:
ð6Þ
Henceforth, considering also an initial condition, system (4) can be rewritten as
@Uðx; tÞ ¼ DDUðx; tÞ þ FðUÞ; x 2 X; @t @U ðx; tÞ ¼ 0; x 2 @ X; t > 0 @g Uðx; 0Þ ¼ uðxÞ ¼ ðu0 ; . . . ; un Þ;
t>0 ð7Þ
x 2 X:
Let X be the Banach space X 0 X n , where X i ¼ CðXÞ; i ¼ 0; . . . ; n. The norm on X is defined by juj ¼ ju0 j þ þ jun j. Let A0i ; i ¼ 0; . . . ; n be the differential operators A0i wi ¼ Di Dwi , defined on the domains DðA0i Þ; i ¼ 0; . . . ; n; where
DðA0i Þ ¼
@wi wi 2 C 2 ðXÞ \ C 1 ðXÞ : A0i wi 2 CðXÞ; ðxÞ ¼ 0; x 2 @ X ; @g
i ¼ 0; . . . ; n:
The closures Ai of A0i in X i generate analytic semigroups of bounded linear operators T i ðtÞ; i ¼ 0; . . . ; n for t P 0 such that wi ðtÞ ¼ T i ðtÞui are solutions of the abstract linear differential equations in X i given by
w0i ðtÞ ¼ Ai wi ðtÞ;
i ¼ 0; . . . ; n:
An additional property of the semigroup is that for each t > 0; T i ðtÞ; i ¼ 0; . . . ; n are compact operators. Let TðtÞ : X ! X be defined by TðtÞ ¼ T 0 ðtÞ T n ðtÞ. Then TðtÞ is a semigroup of operators on X generated by the operator A ¼ A0 An defined on DðAÞ ¼ DðA0 Þ DðAn Þ and Uðx; tÞ ¼ ½TðtÞuðxÞ is the solution of the linear system
@U ðx; tÞ ¼ DDUðx; tÞ; x 2 X; t > 0 @t @U ðx; tÞ ¼ 0; x 2 @ X; t > 0; Uðx; 0Þ ¼ uðxÞ; x 2 X: @g
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C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
In the language of partial differential equations Uðx; tÞ ¼ ½TðtÞuðxÞ are classical solutions of the initial boundary value problem (5), with F ¼ 0. Observe that the nonlinear term F is twice continuously differentiable. Therefore, we can define the map ½F ðuÞðxÞ ¼ FðuðxÞÞ which maps X into itself and equation (5) can be viewed as the abstract ODE in X given by
w0 ðtÞ ¼ AwðtÞ þ F ðwðtÞÞ;
wð0Þ ¼ u;
w ¼ ðw0 ; . . . ; wn Þ:
ð8Þ
While a solution wðtÞ of (6) can be obtained under the restriction that u 2 DðAÞ, a mild solution can be obtained for every u 2 X by requiring only that wðtÞ is a continuous solution of the following integral equation:
wðtÞ ¼ TðtÞu þ
Z
t
Tðt sÞF ðwðsÞÞds;
t 2 ½0; bÞ;
0
where b ¼ bðuÞ 6 1. Restricting our attention to functions u in the set
X K ¼ fu 2 X : uðxÞ 2 K; x 2 Xg; nþ1 where K ¼ Rþ , and taking into account the definition of the functions fi , we obtain that
f0 ð0; v 1 ; . . . ; v n Þ ¼ 0;
f i ðu; v 1 ; . . . ; v i1 ; 0; v iþ1 ; . . . ; v n Þ ¼ 0; i ¼ 1; . . . ; n
for U 2 K. Thus, Corollary 3.2, p. 129 in [10] implies that the Nagumo condition for the positive invariance of K is satisfied, i.e. 1
lim h distðK; U þ hFðUÞÞ ¼ 0;
U 2 K:
h!0þ
ð9Þ
On the other hand, a direct applications of the strong parabolic maximum principle can be used to show that the linear semigroup TðtÞ leaves X K positively invariant, i.e.
TðtÞX K X K ;
t P 0:
ð10Þ
Remark 1. Let us pointing out that the invariance of the set X K can be obtained directly, just having in mind the special structure of the vector field F. Indeed, the coordinate hyperplanes u ¼ 0; v i ¼ 0; i ¼ 1; . . . ; n are invariant under the semiflow induced by the system (1). Now, we are going to prove that all solutions of the system (4) are bounded and therefore defined for all t P 0. Theorem 2.1. (Dissipativeness). Let ðu; v 1 ; . . . ; v n Þ be any solution of (4). Then
lim sup max uðx; tÞ 6 1; t!1
lim sup max v i ðx; tÞ 6 maxf0; t!1
ð11Þ
x2X
x2X
ðbi di Þ g; di
i ¼ 1; . . . ; n:
ð12Þ
Proof. From the first equation of the system (4), it follows that
@u D0 Du 6 uð1 uÞ; @t as long as u is defined as a function of t. Let z be the solution of the equation
z0 ðtÞ ¼ zð1 zÞ; zð0Þ ¼ max uðx; 0Þ: x2X
From the comparison principle (see Theorem 10.1, p. 94 in [11]), we obtain uðx; tÞ 6 zðtÞ. Now, taking into account that for any > 0 there exists a T > 0 such that zðtÞ < 1 þ for any t P T , which in turn implies that uðx; tÞ is defined for all t P 0, and
lim sup max uðx; tÞ 6 1: t!1
x2X
Having in mind that for a given > 0 there exists a T > 0 such that uðx; tÞ 6 1 þ for any x 2 X and t P T , and by using the second equation of (4), we get
@v i u 1þ ð1 þ Þðbi di Þ di v i 6 v i di þ bi ¼ vi Di Dv i ¼ v i di þ bi u þ vi @t 1 þ þ vi 1 þ þ vi for any x 2 X and t P T . If bi > di , let zi be the solution of the following initial value problem
z0i ðtÞ ¼
zi ½ð1 þ Þðbi di Þ di zi ; 1 þ þ zi
zi ðT Þ ¼ max v i ðx; T Þ > 0: x2X
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
101
Then
lim zi ðtÞ ¼
t!þ1
ð1 þ Þðbi di Þ : di
Hence, by using the comparison principle, we obtain that
v i ðx; tÞ 6 zi ðtÞ; which implies that
ðb di Þ lim sup max v i ðx; tÞ 6 i : di t!1 x2X If bi 6 di , we have
di v 2i @v i ; Di Dv i 6 @t 1 þ þ vi which implies that
lim sup max v i ðx; tÞ 6 0; t!1
i ¼ 1; . . . ; n:
x2X
This completes the proof of our claim.
h
Remark 2. From the proof of the dissipativeness, we obtain as a corollary that the predators cannot survive provided that bi 6 di , for i ¼ 1; . . . ; n. In the original variables, this means that the necessary condition for the survival of each predator the birth rate mi have to be larger than the death rate ai , i.e. mi > ai . Remark 3. Another immediate consequence of the proof of the former result is that for a given fu 2 X K : uðxÞ 2 Bg is an absorbing set for the system (4), where
> 0; A ¼
ðb d1 Þ ðb dn Þ B ¼ ½0; 1 þ 0; 1 þ 0; n þ : d1 dn
Finally, Theorem 2.1, conditions (8) and (9) together allow us apply Theorem 3.1, p. 127 in [10], giving us Lemma 2.1. For each u 2 X K , (4) has a unique mild solution wðtÞ ¼ wðu; tÞ 2 X K and a classical solution Uðx; tÞ ¼ ½wðtÞðxÞ. Moreover, the set X K is positively invariant under flow Wt ðuÞ ¼ wðu; tÞ induced by (4). So, the model (4) is biologically well possed. Moreover, from the previous theorem and by using the Theorem 3.4.8, p. 40, in [7], it follows that the relevant dynamic of the system (4) is concentrated on a compact set of the space X K , which is contained in A . 3. Uniform persistence of the predators The goal of this section is to give conditions implying that the predators and prey persist indefinitely, i.e., that neither becomes extinct. Definition 3.1. Problem (4) is said to have the persistence property if, for any non-negative initial data uðxÞ, with ui ðxÞX0; i ¼ 0; . . . ; n; there exists a positive constant e ¼ eðu) such that the corresponding solution ðu; v 1 ; . . . ; v n Þ of (1) satisfies
lim inf min uðx; tÞ P e; t!1 x2X
lim inf min v i ðx; tÞ P e; i ¼ 1; . . . ; n: t!1 x2X
Theorem 3.1. If bi > di ; i ¼ 1; . . . ; n and
Pn
i¼1 si
< 1, then (4) has the persistence property.
Proof. Let us suppose that ui ðxÞ > 0; i ¼ 0; . . . ; n. From the first equation of the system (4), we obtain n n n X X X @u si uv i si þ u2 D0 Du ¼ uð1 uÞ ¼ uð1 uÞ u @t vi þ u i¼1 i¼1 i¼1
Since
Pn
i¼1 si
< 1, it follows by a comparison argument that
lim inf min uðx; tÞ P 1 t!1
X
n X
si > 0:
i¼1
Thus, there exists a T > 0 such that
uðx; tÞ P
" # n X 1 si ¼ g > 0 8x 2 X; t P T: 1 2 i¼1
! n X si si : Pu 1u vi þ u i¼1
102
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
Taking into account this inequality and by using the second equation of (4), we get
@v i u g gðbi di Þ di v i Di Dv i ¼ v i di þ bi P v i di þ bi P vi g þ vi u þ vi @t g þ vi for any x 2 X and t P T. Let zi be the solution of the following initial value problem:
z0i ðtÞ ¼
zi ½gðbi di Þ di zi ; g þ zi
zi ðTÞ ¼ min v i ðx; T Þ > 0: x2X
Then
lim zi ðtÞ ¼
gðbi di Þ di
t!þ1
> 0;
since bi > di . Hence, by using the comparison principle, we obtain that
lim inf min v i ðx; tÞ P
gðbi di Þ
t!1 x2X
di
;
v i ðx; tÞ P zi ðtÞ; which implies that
i ¼ 1; . . . ; n:
Which in turn completes the proof of our assertion.
h
Let us finish this section with a couple of remarks. Remark 4. In the theorem 3.1, it might be possible to relax the assumption u1 ðxÞ > 0 for all x 2 X (which is required so that zð0Þ > 0), to u1 ðxÞ P 0 for all x 2 X with u1 ðxÞX0. In this later situation the strong maximum principle implies u1 ðx; tÞ > 0 for all t > 0; x 2 X. One could change the origin of time to a positive time t1 and still get zð0Þ > 0 for an appropriately modified comparison problem. Remark 5. It is interesting to point out, that under the conditions for the persistence of the system (4), the interaction matrix of system (4) is sign stable, namely the coefficient matrix of the ODE connection with (4) linearized at ðu ; v 1 ; v 2 ; . . . ; v n Þ is sign stable. For more details we refer the reader to Theorem 2.2. in [9]. 4. Global stability of the nontrivial equilibrium The main concern of this section is to prove the global stability of ðu ; v 1 ; . . . ; v n Þ, the unique nontrivial equilibrium of the system (4), where the coordinates of the equilibrium are given by
u ¼ 1
n X si ðbi di Þ ; bi i¼1
v i ¼
ðbi di Þ u ; i ¼ 1; . . . ; n: di
In order to achieve our goal, we will use basically the method of upper and lower solutions combined with the monotone iteration method. More concretely, we are going to construct 2n þ 2 sequences, namely ð1Þ ð2Þ ð1Þ ð2Þ fxk g; fxk g; fyi;k g; fyi;n g for i ¼ 1; . . . ; n; which satisfy the following properties: ð1Þ
ð2Þ
(i) xk 6 lim inf t!1 minX uðx; tÞ 6 lim supt!1 maxX uðx; tÞ 6 xk , (ii) (iii) (iv)
ð1Þ yi;k 6 lim inf t!1 minX i ðx; tÞ 6 lim supt!1 maxX i ðx; tÞ ð1Þ ð1Þ xk ; yi;k ; i ¼ 1; . . . ; n, are non-decreasing sequences. ð2Þ ð2Þ xk ; yi;k i ¼ 1; . . . ; n, are non-increasing sequences.
v
v
ð2Þ
6 yi;k ; i ¼ 1; . . . ; n,
Let us set
x1 ¼ 1
n X
si ;
x2 ¼ 1;
i¼1
yi;1 ¼ g
bi di ; di
yi;2 ¼
bi di ; di
i ¼ 1; . . . ; n:
ð13Þ
From theorems 2.1 and 3.1 we have that the following bounded closed domain
H ¼ fðu; v 1 ; . . . ; v n Þ : ðu; v 1 ; . . . ; v n Þ 2 ½x1 ; x2 ½y1;1 ; y1;2 ½yn;1 ; yn;2 g is eventually invariant for system (4), i.e., for any positive solution ðu; v 1 ; . . . ; v n Þ of the system (4), the following inequalities hold:
1
n X
si 6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 1;
i¼1
g
t!1
X
t!1
bi di b di 6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 i t!1 di di t!1 X X
for i ¼ 1; . . . ; n.
ð14Þ
X
ð15Þ
103
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
Let us start constructing the sequences, recurrently. Let us set ð1Þ
ð2Þ
x1 ¼ x 1 ;
ð1Þ
x1 ¼ x2 ;
ð2Þ
yi;1 ¼ yi;1 ;
yi;1 ¼ yi;2 ;
i ¼ 1; . . . ; n;
where x1 ; x2 ; yi;1 ; yi;2 ; i ¼ 1; . . . ; n, were defined in (13). From (14) and (15), it follows that for any exists a T > 0 such that ð1Þ
e > 0 small enough, there
ð2Þ
x1 e 6 uðx; tÞ 6 x1 þ e ð1Þ
ð2Þ
yi;1 e 6 v i ðx; tÞ 6 yi;1 þ e;
ð16Þ
i ¼ 1; . . . ; n
for any t > T, uniformly for x 2 X. First of all, let us observe that (16) implies that
u
616
ð2Þ
x1 þ e
vi
vi
and
ð1Þ
yi;1 e
ð2Þ
yi;1 þ e
616
u
:
ð1Þ
x1 e
From these inequalities, it is not difficult to obtain the following estimations ð1Þ
yi;1 e ð1Þ
ð2Þ
6
yi;1 þ x1
ð2Þ
vi u þ vi
yi;1 þ e
6
ð2Þ
ð1Þ
yi;1 þ x1
ð17Þ
:
By using the first equation of the system (4), and the estimations (16) and (17), we obtain that
" # ð1Þ n n X X yi;1 e @u uv i si 6u 1 si ð1Þ u ; D0 Du ¼ uð1 uÞ ð2Þ @t u þ vi y þ x1 i¼1 i¼1
t > T; x 2 X;
" # ð2Þ n n X X yi;1 þ e @u uv i si Pu 1 si ð2Þ u ; D0 Du ¼ uð1 uÞ ð1Þ @t u þ vi y þ x1 i¼1 i¼1
t > T; x 2 X:
i;1
and
i;1
Applying comparison techniques to each of the above problems, we certainly obtain that
1
n X
ð2Þ
si
i¼1
yi;1 þ e ð2Þ
6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 1
ð1Þ
t!1
yi;1 þ x1
t!1
X
X
n X
ð1Þ
si
i¼1
yi;1 e ð1Þ
ð2Þ
yi;1 þ x1
:
Now, from the second equation of the system (4), estimation (16) and the comparison principle, we get
ðbi di Þ ð1Þ ðb di Þ ð2Þ ðx1 eÞ 6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 i ðx1 þ eÞ t!1 di di t!1 X X for i ¼ 1; . . . ; n. Since e is arbitrary positive, we may assert that the following inequalities hold:
1
n X
ð2Þ
si
i¼1
yi;1 ð2Þ
6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 1
ð1Þ
t!1
yi;1 þ x1
t!1
X
X
n X i¼1
ð1Þ
si
yi;1 ð1Þ
ð2Þ
yi;1 þ x1
;
ðbi di Þ ð1Þ ðb di Þ ð2Þ x1 6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 i x1 t!1 di di t!1 X X for i ¼ 1; . . . ; n. Now, we are in position to define the second terms of the sequences under construction, setting ð1Þ
x2 ¼ 1
n X
ð2Þ
si
i¼1 ð1Þ
yi;2 ¼
yi;1 ð2Þ yi;1
ðbi di Þ ð1Þ x1 ; di
þ
ð2Þ
ð1Þ x1
; x2 ¼ 1
n X i¼1
ð2Þ
yi;2 ¼
ð1Þ
si
yi;1 ð1Þ yi;1
ðbi di Þ ð2Þ x1 ; di
ð2Þ
þ x1
;
i ¼ 1; . . . ; n:
Repeating the above performed process, we obtain the recurrence to construct the sequences n o ð2Þ yi;k ; i ¼ 1; . . . ; n. Actually, they are defined as follows: ð1Þ
x1 ¼ x 1 ;
ð1Þ
n X
ð2Þ
n X
xk ¼ 1
si
i¼1 ð2Þ
x1 ¼ x 2 ;
xk ¼ 1
i¼1 ð1Þ
yi;1 ¼ yi;1 ; for i ¼ 1; . . . ; n.
ð1Þ
yi;k ¼
ð2Þ yi;k1 ð2Þ ð1Þ yi;k1 þ xk1 ð1Þ
si
n o n o n o ð1Þ ð2Þ ð1Þ xk ; xk ; yi;k , and
yi;k1 ð1Þ
ð2Þ
yi;k1 þ xk1
ðbi di Þ ð1Þ xk ; di
ð2Þ
ð18Þ
;
yi;1 ¼ yi;2 ;
ð2Þ
yi;k ¼
ðbi di Þ ð2Þ xk di
104
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
Remark 6. It is worthwhile to point out that from the constructing method of the sequences, they satisfy automatically the properties (i) and (ii) enumerated above. ð1Þ
ð1Þ
ð2Þ
ð2Þ
Proposition 4.1. The sequences xk ; yi;k are non-decreasing sequences; and, xk ; yi;k are non-increasing sequences. ð1Þ
ð1Þ
ð2Þ
ð2Þ
Proof. The proof we will carry out by induction. From the definition of the first terms of the sequences xk ; yi;k ; xn ; yi;k , without any additional restriction on the parameters, we immediately get ð1Þ
x1 ¼ 1
n X
si 6 1
n X
i¼1 ð2Þ
x2 ¼ 1
ð2Þ
yi;2
yi;1
ð1Þ
ð2Þ
ð1Þ
i¼1
yi;1 þ x1
ð2Þ
6 1 ¼ x1 ;
¼ x2 ;
ð1Þ
n X
si
i¼1 ð1Þ yi;1
ð2Þ
si
yi;1
ð2Þ
ð1Þ
yi;1 þ x1
ðb di Þ ðbi di Þ ð1Þ ð1Þ ¼g i 6 x1 ¼ yi;2 di di ðb di Þ ð2Þ ðbi di Þ ð2Þ ¼ i x1 6 ¼ yi;1 di di
for i ¼ 1; . . . ; n. Let us assume now that our claim is true for m; i.e. ð1Þ
ð2Þ
xm1 6 xð1Þ m ;
ð1Þ
xð2Þ m 6 xm1 ;
ð1Þ
ð2Þ
yi;m1 6 yi;m ;
ð2Þ
yi;m 6 yi;m1 :
After a tedious but straightforward computation, we obtain that
xð1Þ m ¼ 1
n X
ð2Þ
si
i¼1 ð2Þ
xmþ1 ¼ 1
n X
yi;m1 ð2Þ
ð1Þ yi;m
ð1Þ
ð2Þ
yi;mþ1
þ
ð2Þ xm
n X
ð2Þ
si
i¼1
ð1Þ
yi;m
si
i¼1
yi;m ¼
ð1Þ
yi;m1 þ xm1
61
61
n X
yi;m
ð1Þ
ð2Þ
¼ xmþ1 ;
ð1Þ
yi;m þ xm
ð1Þ yi;m1 ð1Þ ð2Þ yi;m1 þ xm1
si
i¼1
¼ xð2Þ m ;
ðbi di Þ ð1Þ ðb di Þ ð1Þ ð1Þ xm1 6 i xm ¼ yi;mþ1 ; di di ðb di Þ ð2Þ ðbi di Þ ð2Þ ð2Þ ¼ i xm 6 xm1 ¼ yi;m di di
for i ¼ 1; . . . ; n. This complete the proof of our assertion.
h
Proposition 4.1 implies that all sequences in (18) are convergent. Let us denote their limits by ð1Þ
ð2Þ
lim xk ¼ xð1Þ ;
lim xk ¼ xð2Þ ;
k!1
k!1
ð1Þ
ð1Þ
ð2Þ
lim yi;k ¼ yi ;
ð2Þ
lim yi;k ¼ yi
k!1
k!1
for i ¼ 1; . . . ; n. So, taking into account the definition of the sequences and letting k ! 1, we obtain that
xð1Þ ¼ 1
n X i¼1
ð1Þ
yi
¼
n X
ð2Þ
si
yi ð2Þ
yi þ xð1Þ
ðbi di Þ ð1Þ x ; di
ð2Þ
yi
;
xð2Þ ¼ 1
ð1Þ
si
i¼1
¼
yi ð1Þ
yi þ xð2Þ
; ð19Þ
ðbi di Þ ð2Þ x di
for i ¼ 1; . . . ; n. Which immediately imply that
x1 6 xð1Þ 6 lim inf min uðx; tÞ 6 lim sup max uðx; tÞ 6 xð2Þ 6 x2 ; t!1
yi;1 6
ð1Þ yi
X
t!1
X ð2Þ
6 lim inf min v i ðx; tÞ 6 lim sup max v i ðx; tÞ 6 yi t!1
X
t!1
X
6 yi;2 ;
i ¼ 1; . . . ; n. Theorem 4.1. If
Pn
i¼1 si
< 1=3, then the unique positive constant equilibrium ðu ; v 1 ; . . . ; v n Þ of system (4) is a global attractor. ð1Þ
Proof. In order to prove this theorem, it is enough to show that xð1Þ ¼ xð2Þ ¼ u , and yi (19), we have,
ð2Þ
¼ yi
¼ v i ; i ¼ 1; . . . ; n. Indeed, from
105
C. Duque et al. / Applied Mathematics and Computation 208 (2009) 98–105
xð1Þ ¼ 1
n X i¼1
x
ð2Þ
¼1
n X i¼1
si ½ðbi di Þ=di xð2Þ ; ½ðbi di Þ=di xð2Þ þ xð1Þ
ð20Þ
si ½ðbi di Þ=di xð1Þ : ½ðbi di Þ=di xð1Þ þ xð2Þ
Let us ni ¼ ðbi di Þ=di , now from (20),
xð1Þ xð2Þ ¼ ðxð1Þ xð2Þ Þ
n X i¼1
ðni
si ni ðxð1Þ þ xð2Þ Þ : þ xð2Þ Þðni xð2Þ þ xð1Þ Þ
xð1Þ
If xð1Þ < xð2Þ , then
1¼
n X i¼1
n n n n X X X X si ni ðxð1Þ þ xð2Þ Þ si ni si ni 2si ni < 6 þ < ð1Þ þ xð2Þ Þ ð2Þ þ xð1Þ Þ ðni xð1Þ þ xð2Þ Þðni xð2Þ þ xð1Þ Þ ðn x ðn x ðn þ 1Þx 1 i i i i¼1 i¼1 i¼1 i¼1
2si < 1; n P 1 si i¼1
P ð1Þ ð2Þ since ni¼1 si < 1=3. Contradiction. So xð1Þ ¼ xð2Þ and together with (19) yi ¼ yi , i ¼ 1; . . . ; n. ð1Þ ð1Þ Finally, by using (19), we obtain that x ¼ u and yi ¼ v i ; i ¼ 1; . . . ; n. This complete the proof. h 5. Discussion and remarks Let us start pointing out that for one predator prey systems with delay but without diffusion Tang et al. in [12] and for one predator prey systems with diffusion but without delay Fan et al. in [6] discussed the dynamic of the model and in particular the global stability of the nontrivial equilibrium. In both cases they considered a functional response of a ratio-dependent type. Later on, Kiss and Kovács in [9] studied the main features of a model of n predators and one prey species without diffusion, there they performed a thorough analysis of the stability of the nontrivial equilibrium. In this paper we considered a model of n predators and one prey species with diffusion and Michaelis–Menten-type ratio-dependent functional response. We were able to identify the region of persistence of the model and obtain conditions that guaranty the global stability of the nontrivial equilibrium of the model. Having in mind [9], it is easy Pn to show that if i¼1 si < 1=3; i.e., when the nontrivial equilibrium of the system (4) is a global attractor, then the interaction matrix of system (4) without diffusion is sign stable, namely the coefficient matrix of the ODE connection with (4) linearized at ðu ; v 1 ; v 2 ; . . . ; v n Þ is sign stable. This is quite natural from the biological and mathematical point of view. A lot questions remain without answer. Actually, the connection between the stability of the unique nontrivial equilibrium for the model with and without diffusion. In other words, patterns may arise for this model. Another very interesting question is the study of the impact of the dynamic that emerges from the trivial equilibrium. Let us remind that the vector field can be extended by continuity to the origin. This allows us to consider the origin as an equilibrium point of the model. All these question will be analyzed in a forthcoming paper. References [1] H.R. Akcakaya, R. Arditi, L.R. Ginzburg, Ratio-dependent prediction: an abstraction that works, Ecology 76 (1995) 995–1004. [2] R. Arditi, A.A. Berryman, The biological paradox, Trends Ecol. Evol. 6 (1991) 32. [3] R. Arditi, L.R. Ginzburg, H.R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent models, Am. Nat. 138 (1991) 1287– 1296. [4] R. Arditi, L.R. Ginzburg, Coupling in predator–prey dynamics: ratio-dependence, J. Theor. Biol. 139 (1989) 311–326. [5] C. Cosner, D.L. DeAngelis, J.S. Ault, D.B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol. 56 (1999) 65–75. [6] Y.H. Fan, W.T. Li, Global asymptotic stability of a ratio-dependent predator–prey system with diffusion, J. Comput. Appl. Math. 188 (2006) 205–227. [7] J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 1988. Number 25. [8] S.B. Hsu, T.W. Hwang, Y. Kuang, Global analysis of the Michaelis–Menten-type ratio-dependent predator–prey system, J. Math. Biol. 42 (2001) 489– 506. [9] K. Kiss, S. Kovács, Qualitative behaviour of n-dimensional ratio-dependent predator–prey systems, Appl. Math. Comput. 199 (2) (2008) 535–546. [10] L. Smith Hal, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, 1995. [11] J. Smoller, Shock Waves and Reaction–Diffusion Equations, second ed., Springer-Verlag, New York, 1994. [12] S. Tang, L. Chen, Global qualitative analysis for a ratio-dependent predator–prey model with delay, J. Math. Anal. Appl. 266 (2002) 401–419.