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A diffusive stage-structured model in a polluted environment
Nonlinear Analysis: Real World Applications 7 (2006) 96 – 108 www.elsevier.com/locate/na
A diffusive stage-structured model in a polluted environment夡 Ling Baia,∗ , Ke Wangb, c a Institute of Mathematics Science, Jilin University, Changchun 130061, China b Department of Mathematics, Northeast Normal University, China c Key Library for Vegetation Ecology, Education Ministry of China, Changchun 130024, China
Received 17 May 2004; accepted 23 November 2004
Abstract In this paper we discuss a spatial nonhomogeneous stage-structured self-diffusion model in the polluted environment. The global stability of the equilibrium can be derived based on the stability of the corresponding ordinary differential system using Liapunov function method. The effect of diffusion on the stability of the system is also investigated. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35K57; 92D25 Keywords: Stage-structured diffusive model; Pollutant; Stability; Liapunov’ function
1. Introduction In recent years, with a rapid pace of industrialization a large amount of pollutants and contaminants released from various industries, motor vehicles, which enter into the environment and affect human, ecosystem and other biological species, so it is absolutely essential for researchers to lay many considerable interests in the study of effects of pollutants on human and ecosystem and the persistence and global stability of the populations 夡 Project supported by the National Science Foundation of China (No. 10171010 and 10201005) and by Major Project of Education Ministry of China (No. 01061).
∗ Corresponding author. Tel.: +8604314638286.
E-mail address: [email protected] (L. Bai). 1468-1218/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2004.11.010
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in polluted environment. Since Hallman and his colleagues proposed a toxicant-population model in the early 1980s [5–7], the authors obtained many interesting and complicated results [11,10,4]. On the other hand, the stage-structured of the species has been considered more and more because in the natural world many species have a life history that take their individual members through two stages: immature and then mature. There are also many excellent works in this fields [1,14,15]. Many observations and experiments have shown that there is a different response to the pollutants in the environment between the mature and immature populations. Therefore, it is necessary to consider the effect of stage structure on population growth in a polluted environment. Meanwhile, in recent years, the role of spatial heterogeneity and dispersal in the dynamics of population has been an important research subject. In fact, in real world, almost all animals not only have stage-structure of immature and mature but also the space of their movement is spatial nonhomogeneous. In view of the above, in this paper, we intend to consider a stage-structure model in a closed polluted region D with smooth boundary jD, the population can diffuse in the area D. It is reasonable to assume that the toxicant have no effect on the mature species. Keeping these in view, the dynamics of stage-structure model assumed to be governed by the following system of partial differential equations: jN1 = N2 − N1 − r1 N1 − N12 − r11 N1 U + D1 ∇ 2 N1 , j t jN2 2 jt = N 1 − r 2 N 2 + D 2 ∇ N 2 , jT = Q − 0 T + U − N1 T + D3 ∇ 2 T , jt jU = −U + T + N T + N . 1 1 0 0 jt
(1.1)
The following initial and boundary conditions on system (1.1) are considered: N1 (x, y, 0) = (x, y) 0, T (x, y, 0) = (x, y) 0,
N2 (x, y, 0) = (x, y) 0, U (x, y, 0) = (x, y) 0,
jN1 jN2 jT jU = = = 0, = jn jn jn jn
(x, y) ∈ D,
t ∈ (0, ∞), (x, y) ∈ jD,
(1.2)
where all coefficients are positive constants and 0 < , 0 < 1; n is the unit outward vector to jD. Q is the rate of introduction of pollutant into the environment beyond the initial concentration, which is assumed to be positive or zero. In model (1.1), N1 (x, y, t) and N2 (x, y, t) are the density (or size, biomass) of the two subpopulations of immature and mature in D at time t and spatial coordinates (x, y) ∈ D, respectively; T (x, y, t) and U (x, y, t) are the concentration of pollutant in the environment and in the body of immature subpopulation; ∇ 2 = j2 /jx 2 + j2 /jy 2 is the Laplace operator, positive constant D1 , D2 and D3 are self-diffusion rate coefficients of N1 (x, y, t), N2 (x, y, t) and T (x, y, t) in D. Suppose the birth rate of the immature subpopulation N1 is proportional to the existing
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mature subpopulation with a proportionality constant (cf. the term N2 in (1.1)); for the immature subpopulation the death rate and transformation rate of mature are proportional to the existing immature subpopulation with proportionality constants r1 and (cf. the terms r1 N1 and N1 ); the immature subpopulation is density restriction (cf. the term N12 in (1.1)); the death rate of the mature population is proportional to the existing mature subpopulation with a proportionality constant r2 (cf. the term r2 N2 in (1.1)); r11 is the depletion rate coefficient of immature subpopulation N1 due to organismal pollutant concentration (cf. the terms r11 N1 U ) the pollutant in the environment is washed out or broken down with rate 0 and fraction 0 of it may again reenter into immature subpopulation with the uptake of pollutant (cf. the terms 0 T and 0 0 T ); is the depletion rate coefficient of the pollutant in the environment due to its take by immature subpopulation (cf. the term N1 T ); is natural depletion rate coefficient of U due to ingestion and depuration of pollutant (cf. the term U ) and fraction of it may again reenter in the environment (cf. the terms U ); is the net uptake of pollutant from resource by immure subpopulation (cf. the term N1 ). Finally, we assume that the change of pollutant in the environment that comes from uptake and ingestion by the mature organisms be neglected. jN1 jN2 jn and jn are exterior normal derivative for boundary jD, the equality (1.2) is called homogeneous Neumann boundary condition whose meaning can be understood as two possibility as follows: (1) the population N1 (x, y, t), N2 (x, y, t) and T (x, y, t) are isolated by D, namely, the population cannot come in or go out from boundary jD, Neumann boundary condition exclude migration across the boundary; (2) the population stays in the dynamic balance, the population is equal passing in and out any location of boundary jD at any time. Model (1.1) governed by partial differential equations describes real evolutionary processes of the population more precise than ordinary differential equation. In this paper, we will discuss the spatially nonhomogeneous stage-structured model (1.1). It will be shown that under some reasonable conditions, if the equilibrium of system (1.1) without diffusion is globally asymptotically stable, then the corresponding uniform steady state of the initial-boundary value problems (1.1)–(1.2) is also globally asymptotically stable.
2. Model without diffusion We first analyze model (1.1) without diffusion, namely, D1 = D2 = D3 = 0. It is obvious that in this case the system can be written as dN 1 = N2 − N1 − r1 N1 − N12 − r11 N1 U, dt dN2 = N 1 − r 2 N 2 , dt dT = Q − 0 T + U − N1 T , dt dU = −U + T + N T + N . 0 0 1 1 dt
(2.1)
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2.1. Existence of equilibria System (2.1) has two possible nonnegative equilibria by solving the following algebraic equations:
N2 − N1 − r1 N1 − N12 − r11 N1 U = 0, N1 − r2 N2 = 0, Q − 0 T + U − N1 T = 0, −U + 0 0 T + N1 T + N1 = 0. Denote
0 Q Q , E0 = 0, 0, 0 (1 − 0 ) (1 − 0 )
(2.2)
and E ∗ = (N1∗ , N2∗ , T ∗ , U ∗ ), where N1∗ , N2∗ , T ∗ , U ∗ satisfy the following equalities: N2 = T =
N1 , r2
Q + N1 =: f (N1 ), (1 − 0 )0 + (1 − )N1
(0 + N1 )f (N1 ) − Q =: g(N1 ). From the expression of g and f, it is easy to know T = f (N1 ) > 0, U = g(N1 ) > 0 when N1 0. From the first and second equalities of (2.2) we know U=
N1 =
− − r1 − r11 U . r2
Let A = r2 − − r1 and F (N1 ) = N1 − A + r11 g(N1 ), obviously, the equilibrium of system (2.1) satisfies F (N1 ) = 0. If A 0, F (N1 ) > 0 for N1 0, therefore system (2.1) has no strict positive equilibrium. If A > 0, then F (0) = r11 g(0) − A = r11
r11 0 Q 0 f (0) − Q −A= − A. (1 − 0 )
We note that if Q<
A(1 − 0 ) , r11 0
F (0) = and
r11 0 Q − A<0 (1 − 0 )
A A = r11 g > 0, F
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which shows the existence of N1∗ in the interval (0, A ). The uniqueness of N1∗ must satisfy the following condition: Q<
0 (1 − 0 ) , (1 − )
since f (N1 ) > 0 is equivalent to F (N1 ) > 0. It is very easy to know that the following conclusion: Lemma 2.1. System (2.1) has unique strict positive equilibrium E ∗ = (N1∗ , N2∗ , T ∗ , U ∗ ) if and only if A > 0, Q<
A(1 − 0 ) r11 0
(2.3)
Q<
0 (1 − 0 ) . (1 − )
(2.4)
and
If A 0, System (2.1) has no strict positive equilibrium. In the order to discuss the permanence of system (2.1), at first we analyze the local stability of the equilibria through computing the Jacobian matrices corresponding to each equilibrium. It is easy to compute that E0 is locally asymptotically stable if A 0. The Jacobian matrix of the equilibrium E ∗ is 0 −r11 N1∗ −r1 − − 2N1∗ − r11 U ∗ −r2 0 0 J (E ∗ ) = . 0 −0 − N1∗ −T ∗ T ∗ + 0 0 + N1∗ − 0 ¯ is The characteristic equation of J (E) [( )2 + M + B][( )2 + C + D] = 0, where M = + 0 + N1∗ ,
B = (0 + N1∗ ) − (0 0 + N1∗ ),
C = r1 + + 2N1∗ + r11 U ∗ + r2 ,
D = (r1 + + 2N1∗ + r11 U ∗ )r2 − .
We know 0 < , 0 < 1 implies B > 0, and noting r11 U ∗ =
N2∗ − r1 − − N1∗ , N1∗
N2∗ =
∗ N , r2 1
obviously, D > 0, according to the Routh–Hurwitz criteria, E ∗ (N1∗ , N2∗ , T ∗ , U ∗ ) is locally asymptotically stable. To show that E ∗ is globally asymptotically stable, we introduce the following lemma which construct a region of attraction for system (2.1).
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Lemma 2.2. The set 4 : 0 N1 p, 0 N2 q, 0 < T + U L} = {(N1 , N2 , T , U ) ∈ R+
is positive invariant set including the equilibrium E ∗ , which attract all solutions initiating in the interior of the positive quadrant, where p, q are some positive constants such that p 2 , and L = Q+l p , l = min{(1 − ), (1 − 0 )0 }. p > N1∗ , q > N2∗ , r2 p < q < (r1 +)p+ Proof. On the N1 − N2 plane, we choose p > N1∗ , furthermore, the following inequalities hold:
(r1 + )p + p 2 p
N1 =p
0 N2 q,
N˙ 2 |
N2 =q
0 N1 p
< q − r1 p − p − p 2 < 0,
p − r2 q < 0,
T˙ + U˙ = Q − (1 − )U − (1 − 0 )0 T + N1 Q + p − l(T + U ), where l = min{(1 − ), (1 − 0 )0 }, by the theory of differential inequality, we know T + U Q+l p =: L. On the other hand, we know N˙ 1 |
N1 =0
N2 ,T ,U >0
> 0,
N˙ 2 |
N2 =0
N1 ,T ,U >0
> 0,
T˙ + U˙ | T =0,U =0 > 0, N1 ,N2 >0
which show that is positive invariant set which attracts all solutions initiating in the positive orthant. Theorem 2.3. Under the conditions of Lemma 2.1, the inequality [4 (L + ) + 1 r11 ]2 < 1 4 , [3 + 4 (0 0 + N1∗ )]2 < 3 4 (0
+ N1∗ )
(2.5) (2.6)
hold, where 1 , 4 are positive constants 1 (0 + N1∗ ) 3 = , (2.7) 2 2 L 2 where L is mentioned as Lemma 2.2. Then the equilibrium E ∗ of system (2.1) is globally asymptotically stable with respect to all solutions initiating in the interior of the positive quadrant. Proof. Consider the positive definite function about E ∗ in the following form: 2
(T − T ∗ )2 Ni ∗ ∗ V (N1 , N2 , T , U ) = i Ni − Ni − Ni ln ∗ + 3 Ni 2 i=1
(U − U ∗ )2 , 2 where i are some undecided positive constants. + 4
(2.8)
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Differentiating V with respect to t along the solutions of system (2.1), noting the equalities
N2∗ − N1∗ − r1 N1∗ − N1∗2 − r11 N1∗ U ∗ = 0,
N1∗ − r2 N2∗ = 0,
we have 2
Ni − Ni∗ dNi dV dT dU + 3 (T − T ∗ ) + 4 (U − U ∗ ) , = i dt (2.1) Ni dt dt dt i=1
N1 − N1∗ (N2 − N1 − r1 N1 − N12 − r11 N1 U ) N1 N2 − N2∗ + 2 (N1 − r2 N2 ) N2 + 3 (T − T ∗ )(Q − 0 T + U − N1 T ) + 4 (U − U ∗ )(−U + 0 0 T + N1 T + N1 ), N∗ N2 = 1 (N1 − N1∗ ) − 2∗ − (N1 − N1∗ ) − r11 (U − U ∗ ) N1 N1 N1∗ N1 ∗ + 3 (T − T ∗ )[−0 (T − T ∗ ) − ∗ + 2 (N2 − N2 ) N2 N2
= 1
+ (U − U ∗ ) − T (N1 − N1∗ ) − N1∗ (T − T ∗ )] + 4 (U − U ∗ )[−(U − U ∗ ) + 0 0 (T − T ∗ ) + T (N1 − N1∗ ) + N1∗ (T − T ∗ ) + (N1 − N1∗ )], N2 − N2∗ N2 ∗ 2 = 1 (N1 − N1 ) − N1∗ (N1 − N1∗ ) N1 N1∗ N1 − N1∗ N1 ∗ 2 + 2 (N2 − N2 ) − N2∗ (N2 − N2∗ ) N2 N2∗
− 1 (N1 − N1∗ )2 − 1 r11 (N1 − N1∗ )(U − U ∗ ) − 3 0 (T − T ∗ )2 + 3 (T − T ∗ )(U − U ∗ ) − 3 T (T − T ∗ )(N1 − N1∗ )
− 3 N1∗ (T − T ∗ )2 − 4 (U − U ∗ )2 + 4 0 0 (U − U ∗ )(T − T ∗ ) + 4 N1∗ (U − U ∗ )(T − T ∗ ), + 4 T (U − U ∗ )(N1 − N1∗ ) + 4 (U − U ∗ )(N1 − N1∗ ), 1 N2 2 N1 = − ∗ (N1 − N1∗ )2 − ∗ (N2 − N2∗ )2 N1 N1 N2 N2 1 2 + ∗ (N1 − N1∗ )(N2 − N2∗ ) + ∗ (N1 − N1∗ )(N2 − N2∗ ) N1 N2 − 1 (N1 − N1∗ )2 − (3 0 + 3 N1∗ )(T − T ∗ )2 − 4 (U − U ∗ )2 + (4 T − 1 r11 + 4 )(N1 − N1∗ )(U − U ∗ ) + (3 + 4 0 0 + 4 N1∗ )(U − U ∗ )(T − T ∗ ) − 3 T (T − T ∗ )(N1 − N1∗ ).
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Therefore, 2 1 N1 N dV 2 = − ∗ (N2 − N2∗ ) − (N1 − N1∗ ) dt (2.1) N1 N2 N1 1 1 − A11 (N1 − N1∗ )2 + A12 (N1 − N1∗ )(T − T ∗ ) − A22 (T − T ∗ )2 2 2 1 1 ∗ 2 ∗ ∗ − A11 (N1 − N1 ) + A13 (N1 − N1 )(U − U ) − A33 (U − U ∗ )2 2 2 1 1 ∗ 2 ∗ ∗ − A22 (T − T ) + A23 (T − T )(U − U ) − A33 (U − U ∗ )2 , 2 2 where taking 1 = 2 r2 , and A11 = 1 ,
A22 = 3 (0 + N1∗ ),
A12 = −3 T ,
A33 = 4 ,
A13 = 4 (T + ) − 1 r11 ,
Similar discussion with [4], sufficient conditions for following inequalities hold:
A23 = 3 + 4 (0 0 + N1∗ ). dV dt
to be negative definite are that the
A212 < A11 A22 ,
(2.9)
A213 < A11 A33 ,
(2.10)
A223 < A22 A33 .
(2.11)
Under the suitable choice constants i , i = 1, 2, 3, 4, such as taking 1 = 1, and 2 = r12 , and i satisfy (2.5)–(2.7), it is obvious that (2.9) automatically satisfied noting Lemma 2.2 and (2.5) ⇒ (2.10), (2.6) ⇒ (2.11). Thus let D1 = {dV /dt = 0} = {N1 = N1∗ , N2 = N2∗ , T = T ∗ , U = U ∗ } = E ∗ , set D1 only has an invariant equilibrium which indicates that dV /dt is negative definite with respect to all solutions initiating in the interior of the positive quadrant of system (2.1), hence by LaSalle invariant theorem on stability [8], we know the unique positive equilibrium E ∗ is globally asymptotically stable. Theorem 2.4. If A 0, then E0 is globally asymptotically stable for system (2.1). Proof. Consider the following positive Liapunov function: V (N1 , N2 , T , U ) = 1 N1 + 2 N2 , where i are some undecided positive constants. Calculating the derivative of V along the solution of system (2.1) we have dV
= ( 1 − 2 r2 )N2 + [ 2 − 1 ( + r1 )]N1 − 1 N12 − 1 r11 N1 U . dt
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Taking 2 = r2 1 , noting that 2 − 1 ( + r1 ) = ( r2 − − r1 )1 < 0 due to A < 0, hence
dV
0, dt while dV
D2 = = 0 = {N1 = N2 = 0, T 0, U 0}, dt
if D2 is an invariant set of system (2.1), by Eq. (2.1), we know
0 Q Q , D2 = E0 = 0, 0, 0 (1 − 0 ) (1 − 0 )
which shows E0 is globally asymptotically stable for A 0. Theorem 2.3 shows that model (2.1) is permanent under the conditions of Lemma 2.1, (2.3)–(2.4) indicates that the rate of introduction of pollutant Q determines the persistence of the population, only for Q is appropriately small, the population can sustain at an appropriate positive equilibrium level; A > 0 means r2 +r > 1, where r2 denote a relative birth rate of 1
is regarded as a relative transformation rate of the immature species. immature species, +r 1
Theorem 2.4 depicts that the population is extinctive if only r2 +r 1. 1
3. Model with diffusion In this section we consider the complete model (1.1) and (1.2) and state the main result of this section in the form of the following theorem. Theorem 3.1. (i) If the equilibrium E ∗ of system (2.1) is globally asymptotically stable, the corresponding uniform steady state of the initial-boundary value problem (1.1)–(1.2) is also globally asymptotically stable. (ii) Even if the equilibrium E ∗ of system (2.1) is unstable, the uniform steady state of the initial-boundary value problem (1.1)–(1.2) can be made stable by increasing self-diffusion coefficients appropriately. Proof. Consider the positive definite function W (N1 (t), N2 (t), T (t), U (t)) =
D
V (N1 , N2 , T , U ) dA,
L. Bai, K. Wang / Nonlinear Analysis: Real World Applications 7 (2006) 96 – 108
105
where V is defined by Eq. (2.8). Differentiating W with respect to t along the solutions of system (1.1), we can get dW jV jN 2 jV jT jV jU jV jN1 dA + + + = jU jt jN2 jt jT jt dt (1.1) D jN1 jt = E(x, y, t) dA D jV 2 jV 2 jV 2 + ∇ T dA D1 ∇ N1 + D 2 ∇ N2 + D 3 jN 1 jN 2 jT D = I1 + I2 ,
(3.1)
where
E(x, y, t) = − − − −
2 N2 1 N1 ∗ ∗ (N1 − N1 ) (N2 − N2 ) − N2 N1 N1∗ 1 1 A11 (N1 − N1∗ )2 + A12 (N1 − N1∗ )(T − T ∗ ) − A22 (T − T ∗ )2 2 2 1 1 ∗ 2 ∗ ∗ A11 (N1 − N1 ) + A13 (N1 − N1 )(U − U ) − A33 (U − U ∗ )2 2 2 1 1 ∗ ∗ ∗ 2 A22 (T − T ) + A23 (T − T )(U − U ) − A33 (U − U ∗ )2 , 2 2
I1 =
D
E(x, y, t) dA
and I2 =
D
D1
jV 2 jV 2 jV 2 ∇ T ∇ N1 + D 2 ∇ N2 + D 3 jN 1 jN 2 jT
dA.
Noting the expression of V, for all points in D, we know
j2 V j2 V j2 V = = = 0, jN1 jN2 jN1 jT jN2 jT
j2 V > 0, jN12
j2 V > 0, jN22
j2 V > 0. jT 2
In the following, we check I2 and determine the sign of each term of I2 . Using Green’s formula in the plane D
F ∇ 2 G dA =
jD
F
jG ds − jn
D
∇F · ∇G dA,
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L. Bai, K. Wang / Nonlinear Analysis: Real World Applications 7 (2006) 96 – 108 jG
where jn is the directional derivative in the direction of the unit outward normal to jD and jV s is the arc length. Let F = jN1 and G = N1 , then jV 2 jV jN1 jV · ∇N1 dA ds − ∇ N1 dA = ∇ jN1 D D jN1 jD jN1 jn jV = − · ∇N1 dA, ∇ j N1 D jN
since jn1 = 0, Furthermore, jV j2 V jN1 j2 V jN1 ∇ = i+ j, jN1 jN12 jx jN12 jy Therefore, D
Similarly, D
Hence, I2 =
jV 2 ∇ N1 dA = − jN 1 jV 2 ∇ N2 dA 0, jN 2
D
D
j2 V jN12
jN1 jx
jN1 jN1 i+ j. jx jy
2
jN1 + jy
2 dA 0.
jV 2 ∇ T dA 0. jT
D
∇N1 =
jV 2 jV 2 jV 2 ∇ T D1 ∇ N1 + D 2 ∇ N2 + D 3 jN 1 jN 2 jT
dA 0.
(3.2)
Thus if E ∗ is globally asymptotically stable in the absence of diffusion, the uniform steady state of system (1.1)–(1.2) must be globally asymptotically stable. Furthermore, we note even if E(x, y, t) > 0 leads to I1 > 0, E ∗ is unstable in the absence of diffusion, but Eq. (3.1) shows that we can increase the diffusion coefficients D1 , D2 and D3 to large enough ensuring dW/dt < 0 even if I1 > 0. This proves the second part of Theorem 3.1. Now we shall prove Theorem 3.1 for a given rectangular region D defined by D = {(x, y) : 0 x m, 0 y n}. Thus I2 can be written as jV 2 jV 2 jV 2 D1 ∇ N1 + D 2 ∇ N2 + D 3 ∇ T dA I2 = jN 1 jN 2 jT D 2 2 2 j V jN1 jN1 = − D1 dA + 2 j x jy D jN1 2 j V jN 2 2 jN2 2 − D2 dA + 2 jx jy D jN2 2 2 2 j V jT jT − D3 dA. + 2 jx jy D jT
(3.3)
L. Bai, K. Wang / Nonlinear Analysis: Real World Applications 7 (2006) 96 – 108
107
From Eq. (3.1) about the expression of W, we get N∗ j2 V = i i2 , 2 jNi Ni Hence
j2 V = 3 . jT 2
i = 1, 2;
jN1 2 I2 = − D1 dA + jy D 2 N2∗ jN 2 2 jN 2 2 − D2 dA + jx jy N22 D 2 jT 3 2 jT − D3 dA 3 + j x jy D 1 N1∗ jN 1 2 jN 1 2 dA + − D1 2 jx jy p D 2 N2∗ jN 2 2 jN 2 2 − D2 2 dA + q jx jy D 2 2 jT jT − D3 3 dA. + jx jy D
1 N1∗ N12
jN 1 jx
2
In addition, we note n m j(N1 − N1∗ ) 2 j(N1 − N1∗ ) 2 jN1 2 dA = dA = dx dy. jx jx jx D D 0 0 Let z =
x m,
then
D
jN1 jx
2 dA =
1 m
n 1 j(N
0
0
− N1∗ ) jz
1
Now using the well-known inequality 1 1 jN1 2 2 dx N12 dx jx 0 0 under an analysis similar to [4], we have
2 jN1 2 dA 2 (N1 − N1∗ )2 dA, jx m D D similarly, D
jN1 jy
2
2 dA 2 n
D
(N1 − N1∗ )2 dA.
2 dz dy.
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L. Bai, K. Wang / Nonlinear Analysis: Real World Applications 7 (2006) 96 – 108
Therefore, 1 N1∗ 2 (m2 + n2 ) (N1 − N1∗ )2 dA p 2 m 2 n2 D 2 N2∗ 2 (m2 + n2 ) − D2 (N2 − N2∗ )2 dA q 2 m 2 n2 D 3 2 (m2 + n2 ) − D3 (T − T ∗ )2 dA, m 2 n2 D
I2 − D 1
which shows I2 0.
For a given rectangular region D, by increasing self-diffusion coefficients sufficiently large, an unstable steady state in the absence of self-diffusion can be made stable. This implies that the self-diffusivity plays the role of stability in our model under consideration. References [1] W.G. Aiello, H.I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biol. 101 (1990) 139–153. [4] B. Dubey, J. Hussain, Modelling the interaction of two biological species in a polluted environment, J. Math. Anal. Appl. 246 (1) (2000) 58–79. [5] T.G. Hallam, C.E. Clark, R.R. Lassiter, Effects of toxicant on populations: a qualitative approach I. Equilibrium environmental exposure, Ecol. Modelling 18 (1983) 291–304. [6] T.G. Hallam, C.E. Clark, R.R. Lassiter, Effects of toxicant on populations: a qualitative approach II. First order kinetics, J. Math. Biol. 18 (1983) 25–37. [7] T.G. Hallam, C.E. Clark, R.R. Lassiter, Effects of toxicant on populations: a qualitative approach III. Environment, food chain patchways, J. Theoret. Biol. 109 (1984) 411–429. [8] J. La Sakke, S. Lefschetz, Stability by Liapunov’w Direct Method with Applications, Academic Press, New York, London, 1961. [10] H.P. Lui, Z. Ma, The threshold of survival for system of two species in a polluted environment, J. Math. Biol. 30 (1991) 49–61. [11] Z. Ma, G. Cui, W. Wang, Persistence and extinctions of a population in a polluted environment, Math. Biosci. 101 (1990) 75–97. [14] W. Wang, L. Chen, A predator–prey system with stage structure for predator, Comput. Math. Appl. 33 (8) (1997) 83–91. [15] X. Zhan, L. Chen, A.U. Neumann, The stage-structured predator–prey model and optimal harvesting policy, Math. Biosci. 168 (2000) 201–210.
A diffusive stage-structured model in a polluted environment夡 Ling Baia,∗ , Ke Wangb, c a Institute of Mathematics Science, Jilin University, Changchun 130061, China b Department of Mathematics, Northeast Normal University, China c Key Library for Vegetation Ecology, Education Ministry of China, Changchun 130024, China
Received 17 May 2004; accepted 23 November 2004
Abstract In this paper we discuss a spatial nonhomogeneous stage-structured self-diffusion model in the polluted environment. The global stability of the equilibrium can be derived based on the stability of the corresponding ordinary differential system using Liapunov function method. The effect of diffusion on the stability of the system is also investigated. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35K57; 92D25 Keywords: Stage-structured diffusive model; Pollutant; Stability; Liapunov’ function
1. Introduction In recent years, with a rapid pace of industrialization a large amount of pollutants and contaminants released from various industries, motor vehicles, which enter into the environment and affect human, ecosystem and other biological species, so it is absolutely essential for researchers to lay many considerable interests in the study of effects of pollutants on human and ecosystem and the persistence and global stability of the populations 夡 Project supported by the National Science Foundation of China (No. 10171010 and 10201005) and by Major Project of Education Ministry of China (No. 01061).
∗ Corresponding author. Tel.: +8604314638286.
E-mail address: [email protected] (L. Bai). 1468-1218/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2004.11.010
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in polluted environment. Since Hallman and his colleagues proposed a toxicant-population model in the early 1980s [5–7], the authors obtained many interesting and complicated results [11,10,4]. On the other hand, the stage-structured of the species has been considered more and more because in the natural world many species have a life history that take their individual members through two stages: immature and then mature. There are also many excellent works in this fields [1,14,15]. Many observations and experiments have shown that there is a different response to the pollutants in the environment between the mature and immature populations. Therefore, it is necessary to consider the effect of stage structure on population growth in a polluted environment. Meanwhile, in recent years, the role of spatial heterogeneity and dispersal in the dynamics of population has been an important research subject. In fact, in real world, almost all animals not only have stage-structure of immature and mature but also the space of their movement is spatial nonhomogeneous. In view of the above, in this paper, we intend to consider a stage-structure model in a closed polluted region D with smooth boundary jD, the population can diffuse in the area D. It is reasonable to assume that the toxicant have no effect on the mature species. Keeping these in view, the dynamics of stage-structure model assumed to be governed by the following system of partial differential equations: jN1 = N2 − N1 − r1 N1 − N12 − r11 N1 U + D1 ∇ 2 N1 , j t jN2 2 jt = N 1 − r 2 N 2 + D 2 ∇ N 2 , jT = Q − 0 T + U − N1 T + D3 ∇ 2 T , jt jU = −U + T + N T + N . 1 1 0 0 jt
(1.1)
The following initial and boundary conditions on system (1.1) are considered: N1 (x, y, 0) = (x, y) 0, T (x, y, 0) = (x, y) 0,
N2 (x, y, 0) = (x, y) 0, U (x, y, 0) = (x, y) 0,
jN1 jN2 jT jU = = = 0, = jn jn jn jn
(x, y) ∈ D,
t ∈ (0, ∞), (x, y) ∈ jD,
(1.2)
where all coefficients are positive constants and 0 < , 0 < 1; n is the unit outward vector to jD. Q is the rate of introduction of pollutant into the environment beyond the initial concentration, which is assumed to be positive or zero. In model (1.1), N1 (x, y, t) and N2 (x, y, t) are the density (or size, biomass) of the two subpopulations of immature and mature in D at time t and spatial coordinates (x, y) ∈ D, respectively; T (x, y, t) and U (x, y, t) are the concentration of pollutant in the environment and in the body of immature subpopulation; ∇ 2 = j2 /jx 2 + j2 /jy 2 is the Laplace operator, positive constant D1 , D2 and D3 are self-diffusion rate coefficients of N1 (x, y, t), N2 (x, y, t) and T (x, y, t) in D. Suppose the birth rate of the immature subpopulation N1 is proportional to the existing
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mature subpopulation with a proportionality constant (cf. the term N2 in (1.1)); for the immature subpopulation the death rate and transformation rate of mature are proportional to the existing immature subpopulation with proportionality constants r1 and (cf. the terms r1 N1 and N1 ); the immature subpopulation is density restriction (cf. the term N12 in (1.1)); the death rate of the mature population is proportional to the existing mature subpopulation with a proportionality constant r2 (cf. the term r2 N2 in (1.1)); r11 is the depletion rate coefficient of immature subpopulation N1 due to organismal pollutant concentration (cf. the terms r11 N1 U ) the pollutant in the environment is washed out or broken down with rate 0 and fraction 0 of it may again reenter into immature subpopulation with the uptake of pollutant (cf. the terms 0 T and 0 0 T ); is the depletion rate coefficient of the pollutant in the environment due to its take by immature subpopulation (cf. the term N1 T ); is natural depletion rate coefficient of U due to ingestion and depuration of pollutant (cf. the term U ) and fraction of it may again reenter in the environment (cf. the terms U ); is the net uptake of pollutant from resource by immure subpopulation (cf. the term N1 ). Finally, we assume that the change of pollutant in the environment that comes from uptake and ingestion by the mature organisms be neglected. jN1 jN2 jn and jn are exterior normal derivative for boundary jD, the equality (1.2) is called homogeneous Neumann boundary condition whose meaning can be understood as two possibility as follows: (1) the population N1 (x, y, t), N2 (x, y, t) and T (x, y, t) are isolated by D, namely, the population cannot come in or go out from boundary jD, Neumann boundary condition exclude migration across the boundary; (2) the population stays in the dynamic balance, the population is equal passing in and out any location of boundary jD at any time. Model (1.1) governed by partial differential equations describes real evolutionary processes of the population more precise than ordinary differential equation. In this paper, we will discuss the spatially nonhomogeneous stage-structured model (1.1). It will be shown that under some reasonable conditions, if the equilibrium of system (1.1) without diffusion is globally asymptotically stable, then the corresponding uniform steady state of the initial-boundary value problems (1.1)–(1.2) is also globally asymptotically stable.
2. Model without diffusion We first analyze model (1.1) without diffusion, namely, D1 = D2 = D3 = 0. It is obvious that in this case the system can be written as dN 1 = N2 − N1 − r1 N1 − N12 − r11 N1 U, dt dN2 = N 1 − r 2 N 2 , dt dT = Q − 0 T + U − N1 T , dt dU = −U + T + N T + N . 0 0 1 1 dt
(2.1)
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2.1. Existence of equilibria System (2.1) has two possible nonnegative equilibria by solving the following algebraic equations:
N2 − N1 − r1 N1 − N12 − r11 N1 U = 0, N1 − r2 N2 = 0, Q − 0 T + U − N1 T = 0, −U + 0 0 T + N1 T + N1 = 0. Denote
0 Q Q , E0 = 0, 0, 0 (1 − 0 ) (1 − 0 )
(2.2)
and E ∗ = (N1∗ , N2∗ , T ∗ , U ∗ ), where N1∗ , N2∗ , T ∗ , U ∗ satisfy the following equalities: N2 = T =
N1 , r2
Q + N1 =: f (N1 ), (1 − 0 )0 + (1 − )N1
(0 + N1 )f (N1 ) − Q =: g(N1 ). From the expression of g and f, it is easy to know T = f (N1 ) > 0, U = g(N1 ) > 0 when N1 0. From the first and second equalities of (2.2) we know U=
N1 =
− − r1 − r11 U . r2
Let A = r2 − − r1 and F (N1 ) = N1 − A + r11 g(N1 ), obviously, the equilibrium of system (2.1) satisfies F (N1 ) = 0. If A 0, F (N1 ) > 0 for N1 0, therefore system (2.1) has no strict positive equilibrium. If A > 0, then F (0) = r11 g(0) − A = r11
r11 0 Q 0 f (0) − Q −A= − A. (1 − 0 )
We note that if Q<
A(1 − 0 ) , r11 0
F (0) = and
r11 0 Q − A<0 (1 − 0 )
A A = r11 g > 0, F
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which shows the existence of N1∗ in the interval (0, A ). The uniqueness of N1∗ must satisfy the following condition: Q<
0 (1 − 0 ) , (1 − )
since f (N1 ) > 0 is equivalent to F (N1 ) > 0. It is very easy to know that the following conclusion: Lemma 2.1. System (2.1) has unique strict positive equilibrium E ∗ = (N1∗ , N2∗ , T ∗ , U ∗ ) if and only if A > 0, Q<
A(1 − 0 ) r11 0
(2.3)
Q<
0 (1 − 0 ) . (1 − )
(2.4)
and
If A 0, System (2.1) has no strict positive equilibrium. In the order to discuss the permanence of system (2.1), at first we analyze the local stability of the equilibria through computing the Jacobian matrices corresponding to each equilibrium. It is easy to compute that E0 is locally asymptotically stable if A 0. The Jacobian matrix of the equilibrium E ∗ is 0 −r11 N1∗ −r1 − − 2N1∗ − r11 U ∗ −r2 0 0 J (E ∗ ) = . 0 −0 − N1∗ −T ∗ T ∗ + 0 0 + N1∗ − 0 ¯ is The characteristic equation of J (E) [( )2 + M + B][( )2 + C + D] = 0, where M = + 0 + N1∗ ,
B = (0 + N1∗ ) − (0 0 + N1∗ ),
C = r1 + + 2N1∗ + r11 U ∗ + r2 ,
D = (r1 + + 2N1∗ + r11 U ∗ )r2 − .
We know 0 < , 0 < 1 implies B > 0, and noting r11 U ∗ =
N2∗ − r1 − − N1∗ , N1∗
N2∗ =
∗ N , r2 1
obviously, D > 0, according to the Routh–Hurwitz criteria, E ∗ (N1∗ , N2∗ , T ∗ , U ∗ ) is locally asymptotically stable. To show that E ∗ is globally asymptotically stable, we introduce the following lemma which construct a region of attraction for system (2.1).
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Lemma 2.2. The set 4 : 0 N1 p, 0 N2 q, 0 < T + U L} = {(N1 , N2 , T , U ) ∈ R+
is positive invariant set including the equilibrium E ∗ , which attract all solutions initiating in the interior of the positive quadrant, where p, q are some positive constants such that p 2 , and L = Q+l p , l = min{(1 − ), (1 − 0 )0 }. p > N1∗ , q > N2∗ , r2 p < q < (r1 +)p+ Proof. On the N1 − N2 plane, we choose p > N1∗ , furthermore, the following inequalities hold:
(r1 + )p + p 2 p
N1 =p
0 N2 q,
N˙ 2 |
N2 =q
0 N1 p
< q − r1 p − p − p 2 < 0,
p − r2 q < 0,
T˙ + U˙ = Q − (1 − )U − (1 − 0 )0 T + N1 Q + p − l(T + U ), where l = min{(1 − ), (1 − 0 )0 }, by the theory of differential inequality, we know T + U Q+l p =: L. On the other hand, we know N˙ 1 |
N1 =0
N2 ,T ,U >0
> 0,
N˙ 2 |
N2 =0
N1 ,T ,U >0
> 0,
T˙ + U˙ | T =0,U =0 > 0, N1 ,N2 >0
which show that is positive invariant set which attracts all solutions initiating in the positive orthant. Theorem 2.3. Under the conditions of Lemma 2.1, the inequality [4 (L + ) + 1 r11 ]2 < 1 4 , [3 + 4 (0 0 + N1∗ )]2 < 3 4 (0
+ N1∗ )
(2.5) (2.6)
hold, where 1 , 4 are positive constants 1 (0 + N1∗ ) 3 = , (2.7) 2 2 L 2 where L is mentioned as Lemma 2.2. Then the equilibrium E ∗ of system (2.1) is globally asymptotically stable with respect to all solutions initiating in the interior of the positive quadrant. Proof. Consider the positive definite function about E ∗ in the following form: 2
(T − T ∗ )2 Ni ∗ ∗ V (N1 , N2 , T , U ) = i Ni − Ni − Ni ln ∗ + 3 Ni 2 i=1
(U − U ∗ )2 , 2 where i are some undecided positive constants. + 4
(2.8)
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Differentiating V with respect to t along the solutions of system (2.1), noting the equalities
N2∗ − N1∗ − r1 N1∗ − N1∗2 − r11 N1∗ U ∗ = 0,
N1∗ − r2 N2∗ = 0,
we have 2
Ni − Ni∗ dNi dV dT dU + 3 (T − T ∗ ) + 4 (U − U ∗ ) , = i dt (2.1) Ni dt dt dt i=1
N1 − N1∗ (N2 − N1 − r1 N1 − N12 − r11 N1 U ) N1 N2 − N2∗ + 2 (N1 − r2 N2 ) N2 + 3 (T − T ∗ )(Q − 0 T + U − N1 T ) + 4 (U − U ∗ )(−U + 0 0 T + N1 T + N1 ), N∗ N2 = 1 (N1 − N1∗ ) − 2∗ − (N1 − N1∗ ) − r11 (U − U ∗ ) N1 N1 N1∗ N1 ∗ + 3 (T − T ∗ )[−0 (T − T ∗ ) − ∗ + 2 (N2 − N2 ) N2 N2
= 1
+ (U − U ∗ ) − T (N1 − N1∗ ) − N1∗ (T − T ∗ )] + 4 (U − U ∗ )[−(U − U ∗ ) + 0 0 (T − T ∗ ) + T (N1 − N1∗ ) + N1∗ (T − T ∗ ) + (N1 − N1∗ )], N2 − N2∗ N2 ∗ 2 = 1 (N1 − N1 ) − N1∗ (N1 − N1∗ ) N1 N1∗ N1 − N1∗ N1 ∗ 2 + 2 (N2 − N2 ) − N2∗ (N2 − N2∗ ) N2 N2∗
− 1 (N1 − N1∗ )2 − 1 r11 (N1 − N1∗ )(U − U ∗ ) − 3 0 (T − T ∗ )2 + 3 (T − T ∗ )(U − U ∗ ) − 3 T (T − T ∗ )(N1 − N1∗ )
− 3 N1∗ (T − T ∗ )2 − 4 (U − U ∗ )2 + 4 0 0 (U − U ∗ )(T − T ∗ ) + 4 N1∗ (U − U ∗ )(T − T ∗ ), + 4 T (U − U ∗ )(N1 − N1∗ ) + 4 (U − U ∗ )(N1 − N1∗ ), 1 N2 2 N1 = − ∗ (N1 − N1∗ )2 − ∗ (N2 − N2∗ )2 N1 N1 N2 N2 1 2 + ∗ (N1 − N1∗ )(N2 − N2∗ ) + ∗ (N1 − N1∗ )(N2 − N2∗ ) N1 N2 − 1 (N1 − N1∗ )2 − (3 0 + 3 N1∗ )(T − T ∗ )2 − 4 (U − U ∗ )2 + (4 T − 1 r11 + 4 )(N1 − N1∗ )(U − U ∗ ) + (3 + 4 0 0 + 4 N1∗ )(U − U ∗ )(T − T ∗ ) − 3 T (T − T ∗ )(N1 − N1∗ ).
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Therefore, 2 1 N1 N dV 2 = − ∗ (N2 − N2∗ ) − (N1 − N1∗ ) dt (2.1) N1 N2 N1 1 1 − A11 (N1 − N1∗ )2 + A12 (N1 − N1∗ )(T − T ∗ ) − A22 (T − T ∗ )2 2 2 1 1 ∗ 2 ∗ ∗ − A11 (N1 − N1 ) + A13 (N1 − N1 )(U − U ) − A33 (U − U ∗ )2 2 2 1 1 ∗ 2 ∗ ∗ − A22 (T − T ) + A23 (T − T )(U − U ) − A33 (U − U ∗ )2 , 2 2 where taking 1 = 2 r2 , and A11 = 1 ,
A22 = 3 (0 + N1∗ ),
A12 = −3 T ,
A33 = 4 ,
A13 = 4 (T + ) − 1 r11 ,
Similar discussion with [4], sufficient conditions for following inequalities hold:
A23 = 3 + 4 (0 0 + N1∗ ). dV dt
to be negative definite are that the
A212 < A11 A22 ,
(2.9)
A213 < A11 A33 ,
(2.10)
A223 < A22 A33 .
(2.11)
Under the suitable choice constants i , i = 1, 2, 3, 4, such as taking 1 = 1, and 2 = r12 , and i satisfy (2.5)–(2.7), it is obvious that (2.9) automatically satisfied noting Lemma 2.2 and (2.5) ⇒ (2.10), (2.6) ⇒ (2.11). Thus let D1 = {dV /dt = 0} = {N1 = N1∗ , N2 = N2∗ , T = T ∗ , U = U ∗ } = E ∗ , set D1 only has an invariant equilibrium which indicates that dV /dt is negative definite with respect to all solutions initiating in the interior of the positive quadrant of system (2.1), hence by LaSalle invariant theorem on stability [8], we know the unique positive equilibrium E ∗ is globally asymptotically stable. Theorem 2.4. If A 0, then E0 is globally asymptotically stable for system (2.1). Proof. Consider the following positive Liapunov function: V (N1 , N2 , T , U ) = 1 N1 + 2 N2 , where i are some undecided positive constants. Calculating the derivative of V along the solution of system (2.1) we have dV
= ( 1 − 2 r2 )N2 + [ 2 − 1 ( + r1 )]N1 − 1 N12 − 1 r11 N1 U . dt
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Taking 2 = r2 1 , noting that 2 − 1 ( + r1 ) = ( r2 − − r1 )1 < 0 due to A < 0, hence
dV
0, dt while dV
D2 = = 0 = {N1 = N2 = 0, T 0, U 0}, dt
if D2 is an invariant set of system (2.1), by Eq. (2.1), we know
0 Q Q , D2 = E0 = 0, 0, 0 (1 − 0 ) (1 − 0 )
which shows E0 is globally asymptotically stable for A 0. Theorem 2.3 shows that model (2.1) is permanent under the conditions of Lemma 2.1, (2.3)–(2.4) indicates that the rate of introduction of pollutant Q determines the persistence of the population, only for Q is appropriately small, the population can sustain at an appropriate positive equilibrium level; A > 0 means r2 +r > 1, where r2 denote a relative birth rate of 1
is regarded as a relative transformation rate of the immature species. immature species, +r 1
Theorem 2.4 depicts that the population is extinctive if only r2 +r 1. 1
3. Model with diffusion In this section we consider the complete model (1.1) and (1.2) and state the main result of this section in the form of the following theorem. Theorem 3.1. (i) If the equilibrium E ∗ of system (2.1) is globally asymptotically stable, the corresponding uniform steady state of the initial-boundary value problem (1.1)–(1.2) is also globally asymptotically stable. (ii) Even if the equilibrium E ∗ of system (2.1) is unstable, the uniform steady state of the initial-boundary value problem (1.1)–(1.2) can be made stable by increasing self-diffusion coefficients appropriately. Proof. Consider the positive definite function W (N1 (t), N2 (t), T (t), U (t)) =
D
V (N1 , N2 , T , U ) dA,
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where V is defined by Eq. (2.8). Differentiating W with respect to t along the solutions of system (1.1), we can get dW jV jN 2 jV jT jV jU jV jN1 dA + + + = jU jt jN2 jt jT jt dt (1.1) D jN1 jt = E(x, y, t) dA D jV 2 jV 2 jV 2 + ∇ T dA D1 ∇ N1 + D 2 ∇ N2 + D 3 jN 1 jN 2 jT D = I1 + I2 ,
(3.1)
where
E(x, y, t) = − − − −
2 N2 1 N1 ∗ ∗ (N1 − N1 ) (N2 − N2 ) − N2 N1 N1∗ 1 1 A11 (N1 − N1∗ )2 + A12 (N1 − N1∗ )(T − T ∗ ) − A22 (T − T ∗ )2 2 2 1 1 ∗ 2 ∗ ∗ A11 (N1 − N1 ) + A13 (N1 − N1 )(U − U ) − A33 (U − U ∗ )2 2 2 1 1 ∗ ∗ ∗ 2 A22 (T − T ) + A23 (T − T )(U − U ) − A33 (U − U ∗ )2 , 2 2
I1 =
D
E(x, y, t) dA
and I2 =
D
D1
jV 2 jV 2 jV 2 ∇ T ∇ N1 + D 2 ∇ N2 + D 3 jN 1 jN 2 jT
dA.
Noting the expression of V, for all points in D, we know
j2 V j2 V j2 V = = = 0, jN1 jN2 jN1 jT jN2 jT
j2 V > 0, jN12
j2 V > 0, jN22
j2 V > 0. jT 2
In the following, we check I2 and determine the sign of each term of I2 . Using Green’s formula in the plane D
F ∇ 2 G dA =
jD
F
jG ds − jn
D
∇F · ∇G dA,
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L. Bai, K. Wang / Nonlinear Analysis: Real World Applications 7 (2006) 96 – 108 jG
where jn is the directional derivative in the direction of the unit outward normal to jD and jV s is the arc length. Let F = jN1 and G = N1 , then jV 2 jV jN1 jV · ∇N1 dA ds − ∇ N1 dA = ∇ jN1 D D jN1 jD jN1 jn jV = − · ∇N1 dA, ∇ j N1 D jN
since jn1 = 0, Furthermore, jV j2 V jN1 j2 V jN1 ∇ = i+ j, jN1 jN12 jx jN12 jy Therefore, D
Similarly, D
Hence, I2 =
jV 2 ∇ N1 dA = − jN 1 jV 2 ∇ N2 dA 0, jN 2
D
D
j2 V jN12
jN1 jx
jN1 jN1 i+ j. jx jy
2
jN1 + jy
2 dA 0.
jV 2 ∇ T dA 0. jT
D
∇N1 =
jV 2 jV 2 jV 2 ∇ T D1 ∇ N1 + D 2 ∇ N2 + D 3 jN 1 jN 2 jT
dA 0.
(3.2)
Thus if E ∗ is globally asymptotically stable in the absence of diffusion, the uniform steady state of system (1.1)–(1.2) must be globally asymptotically stable. Furthermore, we note even if E(x, y, t) > 0 leads to I1 > 0, E ∗ is unstable in the absence of diffusion, but Eq. (3.1) shows that we can increase the diffusion coefficients D1 , D2 and D3 to large enough ensuring dW/dt < 0 even if I1 > 0. This proves the second part of Theorem 3.1. Now we shall prove Theorem 3.1 for a given rectangular region D defined by D = {(x, y) : 0 x m, 0 y n}. Thus I2 can be written as jV 2 jV 2 jV 2 D1 ∇ N1 + D 2 ∇ N2 + D 3 ∇ T dA I2 = jN 1 jN 2 jT D 2 2 2 j V jN1 jN1 = − D1 dA + 2 j x jy D jN1 2 j V jN 2 2 jN2 2 − D2 dA + 2 jx jy D jN2 2 2 2 j V jT jT − D3 dA. + 2 jx jy D jT
(3.3)
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From Eq. (3.1) about the expression of W, we get N∗ j2 V = i i2 , 2 jNi Ni Hence
j2 V = 3 . jT 2
i = 1, 2;
jN1 2 I2 = − D1 dA + jy D 2 N2∗ jN 2 2 jN 2 2 − D2 dA + jx jy N22 D 2 jT 3 2 jT − D3 dA 3 + j x jy D 1 N1∗ jN 1 2 jN 1 2 dA + − D1 2 jx jy p D 2 N2∗ jN 2 2 jN 2 2 − D2 2 dA + q jx jy D 2 2 jT jT − D3 3 dA. + jx jy D
1 N1∗ N12
jN 1 jx
2
In addition, we note n m j(N1 − N1∗ ) 2 j(N1 − N1∗ ) 2 jN1 2 dA = dA = dx dy. jx jx jx D D 0 0 Let z =
x m,
then
D
jN1 jx
2 dA =
1 m
n 1 j(N
0
0
− N1∗ ) jz
1
Now using the well-known inequality 1 1 jN1 2 2 dx N12 dx jx 0 0 under an analysis similar to [4], we have
2 jN1 2 dA 2 (N1 − N1∗ )2 dA, jx m D D similarly, D
jN1 jy
2
2 dA 2 n
D
(N1 − N1∗ )2 dA.
2 dz dy.
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L. Bai, K. Wang / Nonlinear Analysis: Real World Applications 7 (2006) 96 – 108
Therefore, 1 N1∗ 2 (m2 + n2 ) (N1 − N1∗ )2 dA p 2 m 2 n2 D 2 N2∗ 2 (m2 + n2 ) − D2 (N2 − N2∗ )2 dA q 2 m 2 n2 D 3 2 (m2 + n2 ) − D3 (T − T ∗ )2 dA, m 2 n2 D
I2 − D 1
which shows I2 0.
For a given rectangular region D, by increasing self-diffusion coefficients sufficiently large, an unstable steady state in the absence of self-diffusion can be made stable. This implies that the self-diffusivity plays the role of stability in our model under consideration. References [1] W.G. Aiello, H.I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biol. 101 (1990) 139–153. [4] B. Dubey, J. Hussain, Modelling the interaction of two biological species in a polluted environment, J. Math. Anal. Appl. 246 (1) (2000) 58–79. [5] T.G. Hallam, C.E. Clark, R.R. Lassiter, Effects of toxicant on populations: a qualitative approach I. Equilibrium environmental exposure, Ecol. Modelling 18 (1983) 291–304. [6] T.G. Hallam, C.E. Clark, R.R. Lassiter, Effects of toxicant on populations: a qualitative approach II. First order kinetics, J. Math. Biol. 18 (1983) 25–37. [7] T.G. Hallam, C.E. Clark, R.R. Lassiter, Effects of toxicant on populations: a qualitative approach III. Environment, food chain patchways, J. Theoret. Biol. 109 (1984) 411–429. [8] J. La Sakke, S. Lefschetz, Stability by Liapunov’w Direct Method with Applications, Academic Press, New York, London, 1961. [10] H.P. Lui, Z. Ma, The threshold of survival for system of two species in a polluted environment, J. Math. Biol. 30 (1991) 49–61. [11] Z. Ma, G. Cui, W. Wang, Persistence and extinctions of a population in a polluted environment, Math. Biosci. 101 (1990) 75–97. [14] W. Wang, L. Chen, A predator–prey system with stage structure for predator, Comput. Math. Appl. 33 (8) (1997) 83–91. [15] X. Zhan, L. Chen, A.U. Neumann, The stage-structured predator–prey model and optimal harvesting policy, Math. Biosci. 168 (2000) 201–210.