文献四：Traveling Wave Solutions for a Class of Predator–Prey Systems

J Dyn Diff Equat(2012)24:633–644
DOI10.1007/s10884-012-9255-4
Traveling Wave Solutions for a Class of Predator–Prey Systems
Wenzhang Huang
Received:30November2011/Revised:29April2012/Published online:22May2012
©Springer Science+Business Media,LLC2012
Abstract We use a shooting method to show the existence of traveling wave fronts and to obtain an explicit expression of minimum wave speed for a class of diffusive predator–prey systems.The existence of traveling wave fronts indicates the existence of a transition zone from a boundary equilibrium to a co-existence steady state and the minimum wave speed measures the asymptotic speed of population spread in some sense.Our approach is a sig-nificant improvement of techniques introduced by Dunbar.The advantage of our method is that it does not need the notion of Wazewski’s set and LaSalle’s invariance principle used in Dunbar’s approach.In our approach,we convert the equations for traveling wave solutions to a system offirst order equations by a“non-traditional transformation”.With this converted new system,we are able to construct a Liapunov function,which gives an immediate impli-cation of the boundedness and convergence of the relevant class of heteroclinic orbits.Our method provides a more efficient way to study the existence of traveling wave solutions for general predator–prey systems.
Keywords Diffusive predator–prey models·Reaction-diffusion equations·Traveling wave solutions·Minimum wave speed
1Introduction
The dynamics of predator–prey interaction is an important subject of the theoretical ecology. It is obvious that one important component of predator–prey interaction is spatial variation in the populations if predators and their prey are spatially distributed.A general class of predator–prey systems,when the spatial variation is a result of diffusion/random movement, can be formulated by a system of reaction-diffusion equations as follows.Let u(x,t)and v(x,t)be the population densities of prey and predator species at a location x∈I R n and time t.Then the dynamical interaction of u(x,t)and v(x,t)can be described by the following system
W.Huang(B)
Department of Mathematical Sciences,University of Alabama in Huntsville,Huntsville,AL35899,USA
e-mail:huangw@
∂u ∂t =d 1 u +B (u )−f (u )v,∂v ∂t =d 2 v +(βf (u )−μ)v,(1.1)
where B (u )is the growth rate of the prey,f (u )is the uptake/functional response to the pred-ator population,the constant βis a conversion rate,μis the death rate of predator species, = n i =1∂2/∂x 2i is the Laplace operator,and d 1and d 2are diffusive rates for prey and predator,respectively.We refer interested readers to the references [14]and [15]for more detailed formulation and biological application of the system (1.1).
The purpose of this paper is to study the existence of wave fronts of (1.1),i.e.,traveling wave solutions connecting two equilibrium points.The importance of existence of traveling wave solutions for a predator–prey system is well recognized.Traveling wave fronts provide target formations and distributions of interacting species,and the asymptotic speed of con-verges of population towards a stable steady state.One can find the detailed accounts on the important application of traveling wave solutions in [4,14,17].
To address more specifically the existence of traveling wave solutions,we need to consider and have certain assumptions on the corresponding reaction equation of (1.1),which is given by a system of ODEs:du dt =B (u )−f (u )v,d v dt
=(βf (u )−μ)v.(1.2)First we suppose that,in the absence of predator,the prey population will converge to a constant K >0,which is known as the environmental carrying capacity.That is,the growth function B (u )for prey population satisfies the condition
A1B (0)=B (K )=0,B (u )>0for u ∈(0,K )and B (u )<0for u >K .
A1implies that E K =(K ,0)is a trivial,boundary equilibrium point of (1.2),an equilibrium state with the absence of predator species.In addition we suppose that (1.2)has a co-existence (positive)equilibrium,for otherwise the predator species would go to extinct and dynamics of (1.2)would be trivial.In this paper,we consider the case that a co-existence equilibrium of (1.2)is unique.That is,the functional response f satisfies the condition
A2f (0)=0,f (u )>0for u >0.Moreover,there is a u ∗∈(0,K )such that βf (u ∗)=μ
and
f (u )<μβ,u ∈[0,u ∗),f (u )>μβ
,u >u ∗.The assumption A2implies that (1.2)has a unique co-existence equilibrium
E ∗=(u ∗,v ∗)with v ∗=B (u ∗)f (u ∗)
.Our interest is to investigate the existence of wave fronts,or traveling wave solutions of (1.1)connecting the equilibria E K and E ∗,i.e.,the existence of solution of (1.1)of the form
u (x ,t )=U (x ·ν+ct ),
v(x ,t )=V (x ·ν+ct )(1.3)
satisfying the boundary condition at infinite:
(U (−∞),V (−∞))=E K ,(U (∞),V (∞))=E ∗.(1.4)
Hereν∈I R n is a unit vector denoting the direction of wave propagation and c∈I R is a wave speed.The existence of solutions of form(1.3)and(1.4)indicates the existence of a transition zone between the equilibria E K and E∗.This presents an important phenomenon in evolutionary ecology,“a species may establish itself in a habitat in which the species was originally absent”[2].In addition,the wave speed,in particular the minimum wave speed of the wave front(1.3)and(1.4),will provide a measure of an very important quantity-the asymptotic spreading speed of population,which shows how quickly the population con-verges to the co-existence equilibrium E∗.[1,7,8,10]and[17]provided a detailed study of the relation between the minimum wave speed and asymptotic spreading speed.
In order to have traveling waves connecting E K and E∗,in addition to assumptions A1 and A2,we also require the stability property on the equilibrium E∗.A sufficient stability condition on E∗can be given as follows.
A3The function g(u)=B(u)
f(u)
has the following property:
g(u)>g(u∗)for u∈[0,u∗)and g(u)<g(u∗)for u∈(u∗,K].
If the functional response f(u)is increasing,which is the case in many applications,such as the Holling Types I,II and III functional responses,with f(0)=0,then A2holds if and only ifβf(K)>μ.A particular example of(1.2)that satisfies all conditions A1–A3is the classical Lotka-V olterra predator–prey model with the logistic growth rate B and Holling type I functional response f.That is,a particular case of(1.2)in which
B(u)=ru
1−
1
K
,
f(u)=au
(1.5)
whenβaK>μ.
The existence of a traveling wave of form(1.3)and(1.4)wasfirst proved by Dunbar[2] in1983with the diffusion coefficient d1=0and functions B and f given by(1.5).In a later paper,Dunbar[3]further extended his work to the case of d1>0.One of key steps in Dunbar’s approach is the construction of a Wazewski set W such that any solution that stays in W for all positive time must be bounded.Thus the LaSalle’s invariance principle can be applied.Moreover,the set W given by Dunbar does not seem simple and is unbounded,and to show the boundedness of a solution staying W for all positive time is a non-trivial task. With d1=0and logistic growth function B(u),Li and Wu[9]used Dunbar’s approach to show the existence of traveling wave solutions for
f(u)=
au2
1+u2
,
a simplified Holling type III functional response when A2and A3are satisfied.Recently Lin et al.[11]extended the result in[9]to a more general Holling type III functional response
f(u)=
au2
1+bu+u
under Assumption A3.[11]made an improvement of Dunbar’s approach by constructing a bounded Wazewski set W so that boundedness of a solution in W becomes automatic. However,the construction of the set W in[11]is complicated and it is unclear whether the
construction can be applied to more general functional responses f(u).For d1>0,J.Huang, G.Lu,and S.Ruan[5]extended Dunbar’s work to a Holling type II functional response
f(u)=
au 1+bu
with some further restriction on parameters,in addition to Assumptions A1–A3.
There are also some results in literature,obtained by using the Conley index,on the existence of traveling wave solutions for abstract predator–prey systems(see reference[13]).
The purpose of this paper is to show the existence of traveling wave solutions of(1.1)for general growth function B(u)and the functional response f(u)under conditions A1–A3. We emphasize here that we do not require the function f to be monotone increasing.It does occur in application in which the functional response f is non-monototic.For example,in
a plant-herbivore model with toxin-determined functional response,formulated recently in
[12],the functional response f is not monotone increasing.Although the central idea of our method is similar to Dunbar’s[2,3],the detailed technique is different from the one given in[2].Wefirst transform the system of second order equations for traveling wave solutions to a system offirst order equations by a particular type of change of variables.This trans-formation enables us to handle more general functions B(u)and f(u)with a much simpler and straightforward way.First,we do not need the notion of Wazewski’s set and do not need the use of LaSalle’s invariance principle used in[2,3].Secondly,for the converted system we can construct a Liapunov function that can simultaneously show both the boundedness and convergence of a relevant class of solutions,which was done in[2,3]by a nontrivial argument.Our technique,which is a significant simplification of Dunbar’s approach,will be more efficient to handle more general birth function B(u)and functional response f(u).
To better illustrate our approach geometrically,we shall in this paper consider the case of d1=0.From the point view of biology d1=0is also justified for some situations.For instance,in many predator–prey interaction models the plant serves as a prey species whose motion,so that whose diffusion effect,is clearly negligible.The technique presented in this paper will be further developed to study the existence of traveling waves d1>0in a separated paper.However,results for d1=0can not be obtained easily,or by a straightforward way from d1>0,by letting d1→0.This gives an additional justification for us to study the case d1=0and d1>0separately.
This paper is organized as follows.Wefirst establish some preliminary results in Sect.2. In Sect.3we will give a complete proof of our main theorem stated below:
Theorem1.1Suppose that assumptions A1–A3are satisfied.In addition suppose d1=0 and let d2=d in(1.1).Then
(a)(1.1)has no nonnegative traveling wave solution connecting E K and E∗if
0<c<2
d
βf(K)−μ
.(1.6)
(b)For each
c≥2
d
βmax{f(u):u∈(u∗,K]}−μ
,(1.7)
(1.1)has a nonnegative traveling wave solution connecting E K and E∗.
(c)If f(K)≥f(u)for u∈[0,K],in particular,if f(u)is monotone in[0,K],then
(1.1)has a nonnegative traveling wave solution connecting E K and E∗if and only if
c ≥2
d βf (K )−μ .That is,
c ∗=2
d βf (K )−μ
is a minimal wave speed.
Finally,a short discussion is given Sect.4.
2Preliminaries
Let d 1=0and d 2=d in (1.1).A direct computation shows that (U (ξ),V (ξ))with ξ=x +ct is a traveling wave solution defined in (1.3)if and only if (U (ξ),V (ξ))is a solution of the system
c ˙U =f (U )[g (U )−V ],c ˙V =
d ¨V +[β[f (U )−μ]V ,(2.1)
where ˙W
and ¨W denote the first and second order derivative of a function W ,respectively.Instead of using a standard transformation,for a constant c >0we make the following changes of variables and scaling:
X (t )=U (ct ),
Y (t )=V (ct ),Z (t )=1c
cV (ct )−d ˙V (ct ) .(2.2)Then,upon a straightforward computation,(2.1)is transformed to a three dimensional system ˙X
=f (X ) g (X )−Y ,˙Y =c 2d Y −Z ,˙Z
= βf (X )−μ Y .(2.3)It is clear that (U (ξ),V (ξ))is a nonnegative solution of (2.1)connecting the equilibrium E K and E ∗if and only if (X (t ),Y (t ),Z (t )),with X (t )≥0,Y (t )≥0,is a solution of (2.3)satisfying the boundary condition
(X (−∞),Y (−∞),Z (−∞))=(K ,0,0)=E K ,(X (∞),Y (∞),Z (∞))=(u ∗,v ∗,v ∗)=E ∗.(2.4)
Here,for convenience,we use the same notations E K and E ∗to denote the equilibrium points (K ,0,0)and (u ∗,v ∗,v ∗)of the system (2.3).
Let
f m =max {f (X ):X ∈[0,K ]}.
Suppose that (X (t ),Y (t ),Z (t )),with 0≤X (t )≤K ,0≤Y (t )is a solution of (2.3).Then we have −βf (u ∗)Y (t )≤˙Z
(t )≤ βf m −μ Y (t ).(2.5)Let σ>1be a constant defined by
σ=c 2+ c 4+4c 2d μ2c 2.(2.6)
Fig.1a In the boundaries P1and P2the vectorfields point out side of the region ,while the vectorfields in the boundary P4point inside of the region ,the boundary P3is invariant.b The line Z=12Y is the projection of boundary P1onto the Y–Z plane,and the line Z=σY is the projection of boundary P2onto the Y–Z plane
We define a wedged region ∈I R3as follows(see Fig.1a).
=
(X,Y,Z):0≤X≤K,Y≥0,1
2
Y≤Z≤σY
.(2.7)
Then the boundary of consists of four pieces P i,for i=1,···,4,with
P1=
0<X<K,Y>0,
1
2
Y=Z
,
P2=
0<X<K,Y>0,Z=σY
,
P3=
X=0,Y>0,
1
2
Y≤Z≤σY
,
P4=
X=K,Y>0,
1
2
Y≤Z≤σY
.
(2.8)
and a line segment
l=
0≤X≤K,Y=Z=0
.
The vectorfield of(2.3)has a very simple property in the boundary of ,which can be characterized by the following two lemmas.
Lemma2.1Let t(p)be theflow of(2.3),i.e. t(p)is a solution of(2.3)satisfying the initial condition 0(p)=p∈I R3.Then for any p∈interior of , t(p)can not leave from a point in the boundary P3∪P4∪l of at any positive time.
Proof It is obvious that the plane{X=0}and the line{Y=Z=0}are invariant sets of (2.3).Hence any solution of(2.3)through a point in the interior of can not leave from a point in the set P3∪l.Moreover,f(K)>0and g(K)=B(K)/f(K)=0by Assumptions A1and A2.It follows that at each point p=(X,Y,Z)∈P4,where X=K and Y>0,
˙X=f(K)[g(K)−Y]<0.
That is,the vector field of (2.3)at each point in the set P 4points into the left-side,the region in which the X -coordinate is less than K ,of the plane X =K .Since the set is in the left-side of the plane X =K ,we therefore conclude that any solution t (p )of (2.3)through a point p in the interior of can not leave from a point in the set P 4at a positive time t . Next let us study the vector field at the sets P 1and P 2.
Lemma 2.2If c >2 d βf m −μ ,then the vector field of (2.3)at any p ∈P 1∪P 2points
to the exterior of .
Proof First consider a point p 1=(X 1,Y 1,Z 1)∈P 1.Then 0<X 1<K and Z 1=12Y 1>0.At p 1,we have
˙Y =c 2d (Y 1−Z 1)=c 2Y 12d >0,˙Z ˙Y =2d βf (X 1)−μ c 2≤2d βf m −μ c 2<12.(2.9)
(2.9)implies that the vector field of (2.3)at p 1points to exterior of (see Fig.1b).Next let p 2=(X 2,Y 2,Z 2)∈P 2.The 0<X 2<K and Z 2=σY 2.With the use of (2.6),we obtain
˙Y =c 2d (Y 2−Z 2)=c 2d (1−σ)Y 2=− c 4+4c 2d μ−c 22d Y 2<0,˙Z ˙Y =2d μ−βf (X 2) 4+4c 2μ−c 2<2d μ 4+4c 2−c
2=c 2+ c 4+4c 2d μ2c =σ.(2.10)(2.10)implies that the vector field of (2.3)at p 2points to exterior of (see Fig.1b).Now we turn to study the unstable manifold of the equilibrium E K .The linearized system at E K is
˙x =B (K )x −f (K )y ,˙y =c 2d y −z ,˙z = βf (K )−μ y .(2.11)
Upon a direct computation,one is able to verify that the linear system (2.11)has one eigen-value λ0=B (K )and two positive eigenvalues λ1=c 2+ c 4−c 2d [βf (K )−μ]2d ,λ2=c 2− c 4−c 2d [βf (K )−μ]2d
,(2.12)with the corresponding eigenvectors to λ1and λ2are
h 1= f (K )B (K )−λ1,1,d λ2c 2 T
,h 2= f (K )B (K )−λ2,1,d λ1c 2 T
.(2.13)
Moreover,Assumption A1implies that B (K )≤0.Hence the unstable manifold E u K of E K
is tangent to the plane
P ={k 1h 1+k 2h 2+E K :k 1,k 2∈I R }
at E K .A further computation shows that P = θ(k 1,k 2),k 1+k 2,d c
2[λ1k 2+λ2k 1] :k 1,k 2∈I R with
θ(k 1,k 2)=K +f (K )[(k 1+k 2)B (K )−(λ1k 2+λ2k 1)](B (K )−λ1)(B (K )−λ2)
.Lemma 2.3There are two points p ∗1∈P 1and p ∗2∈P 2and a curve γ∈E u K which connects p ∗1and p ∗2such that
γ\{p ∗1,p ∗2}∈interior of .
Proof Note that the transform
Y =k 1+k 2,Z =d c 2[λ2k 1+λ1k 2](2.14)
from I R 2→I R 2is invertible.Thus the plane P can also be expressed as
P = K +f (K )[d B (K )Y −c 2Z ]d (B (K )−λ1)(B (K )−λ2),Y ,Z ,Y ,Z ∈I R .(2.15)
Since the unstable manifold E u K is tangent to P at E K ,there is a k >0such that in a small neighborhood of the equilibrium E K the unstable manifold E u K can be expressed as E u K = (X (Y ,Z ),Y ,Z ):(Y ,Z )∈[−k ,k ]2
(2.16)where
X (Y ,Z )=K +
f (K )[d B (K )Y −c 2Z ]d (B (K )−λ1)(B (K )−λ2)+η(Y ,Z )(2.17)
and
η(Y ,Z )=0(Y 2+Z 2)as (Y ,Z )→(0,0).(2.18)Pick a sufficiently small >0with σ ≤k .Recall that B (K )≤0,f (K )>0and λi >0for i =1,2.It follows from (2.17)and (2.18)that
0<X ( ,Z )<K ,for Z ∈[12
,σ ].(2.19)
Now we let γ∈E u K be defined as γ= (X ( ,Z ), ,Z ):Z ∈[12
,σ ] ,and let p ∗1=(X ( , /2), , /2)and p ∗2=(X ( ,σ ), ,σ ).Then it is clear that p ∗i ∈P i for i =1,2and γis a curve connecting p ∗1and p ∗2.Moreover it is obvious that γ\{p ∗1,p ∗2}∈ by (2.19).
3Proof of Theorem 1.1
We shall give a complete proof of Theorem 1.1in this section.Let us first present a sufficient condition for the existence of traveling wave solutions connecting E K and E ∗.
Proposition 3.1Suppose Assumptions A1–A3are satisfied.Then (1.1)has a traveling wave solution connecting E K and E ∗for each c ≥2 d βf m −μ .
Proof It is sufficient to show that Proposition 3.1holds for all c >2 d βf m −μ .The case of c =2 d βf m −μ can be treated by letting c 2 d βf m −μ (for the validity
of such a limiting argument we refer readers to the proof of Theorem 4.1in [6]or the proof of Theorem 4.2in [16]).Let γ⊂E u K ∩ be defined as in Lemma 2.3.We define subsets γ1and γ2of γas follows:
γi ={p ∈γ:there is a t ≥0such that t (p )∈P i },i =1,2.(3.1)
Apparently γi is nonempty since p ∗i ∈γi for i =1,2.Noting that,by Lemma 2.2,the vector field of (2.3)at each point in the plane P 1or P 2points to the exterior of ,with the use of continuity of solutions on the initial condition we easily deduce that both the sets γ1and γ2are open relative to the curve γ.It is obvious that γis a connected set.It therefore follows that
γ\(γ1∪γ2)=∅.
Let p ∗∈γ\(γ1∪γ2).Then the definitions of γ,γi and Lemma 2.1imply that
t (p ∗)∈Interior of for all t ≥0.(3.2)
Let the solution t (p ∗)of (2.3)be given by t (p ∗)=(X (t ),Y (t ),Z (t ))and let L :[0,∞)→I R by defined by
L (t )=X (t ) u ∗1f (s ) f (s )−f (u ∗) ds +1β Z (t )−g (u ∗)Z (t )Y (t )−g (u ∗)ln Y (t ) .(3.3)
Then,by (3.2),L (t )is well defined.Upon a straightforward computation and with the use of the system (2.3)we obtain the derivative of L (t )as
˙L (t )= f (X (t ))−f (u ∗) [g (X (t ))−g (u ∗)]−c 2d g (u ∗)Y 2(t )
Y (t )−Z (t ) 2.(3.4)From Assumptions A2,A3and (3.4)it follows that
˙L
(t )≤0for t ≥0.(3.5)Hence L (t )is decreasing.Moreover (X (t ),Y (t ),Z (t ))∈ yields that
Z (t )Y (t )
≤σ,Z (t )≥12Y (t ).(3.6)From (3.3)and (3.6)if follows that
L (0)≥L (t )≥1β Y (t )2−g (u ∗)σ−g (u ∗)ln Y (t ) .(3.7)
(3.7)implies that
L (0)+g (u ∗)σ≥1β
Y (t )2−g (u ∗)ln Y (t ) .(3.8)
One can easily verify that
Y (t )/2−g (u ∗)ln Y (t )→+∞if Y (t ) 0or Y (t )→+∞.(3.9)From (3.6),(3.8)and (3.9)we conclude that there are positive constants M 1and M 2such that
M 1≤Y (t )≤M 2,M 1≤Z (t )≤M 2,t ≥0.(3.10)
The implication of the inequality (3.10)and express (3.4)is that
(i)
(X (t ),Y (t ),Z (t ))is bounded;(ii)
The function L (t )is monotone decreasing and is bounded below as t →∞;(iii)˙L
(t )is uniformly continuous.From (ii)and (iii)it therefore follows that ˙L
(t )→0as t →∞.So that,with the use of the expression (3.4)again,we obtain
X (t )→u ∗and Y (t )−Z (t )→0as t →∞.
(3.11)Notice that ˙X
(t )is uniformly continuous.Hence X (t )→u ∗yields that ˙X
(t )→0as t →∞.(3.12)
An immediate consequence of (2.3),(3.11),and (3.12)is that
(X (t ),Y (t ),Z (t ))→(u ∗,g (u ∗),g (u ∗))=(u ∗,v ∗,v ∗)=E ∗as t →∞.
Moreover,we have
t (p ∗)=(X (t ),Y (t ),Z (t ))→E K as t →−∞,
for p ∗is in the unstable manifold E u K of E K .Proof of Theorem 1.1First we notice that,for the system (2.3),the subset {(X ,0,0):X >0}is in the stable manifold of E K corresponding to the eigenvalue λ0=B (K )≤0.Hence if (2.3)has a solution t (p )=(X (t ),Y (t ),Z (t ))→E K as t →−∞with X (t )>0and Y (t )>0,then p must be in the unstable manifold of E K corresponding to the eigenvalues λ1and λ2given in (2.12).However,if 0<c <2 d βf (K )−μ ,then eigenvalues λ1and λ2are complex.By looking at the associated eigenvectors given in (2.13),one is able to conclude that if a solution is in the unstable manifold of the equilibrium E K ,then its Y (t )component can not keep nonnegative all the time when the solution converges to E K as t →−∞.Hence Part (a)of Theorem 1.1holds.Part (b)of Theorem 1.1is precisely the Proposition 3.1.Finally,Part (c)is just an immediate consequence of Parts (a)and (b)when f (K )=max {f (u ):u ∈[0,K ]}.
4Discussion
We have shown that,if the functional response f(u)has the property that f(K)=max{f(u): u∈[0,K]},then we have an explicit and simple expression for the minimum wave speed
c∗=2
d
βf(K)−μ
.(4.1)
This actually is a so-called linear determinacy situation.That is,the minimum wave speed is determined by the eigenvalues of linearization of(2.3)at the unstable equilibrium point E K (see the proof of Part(a)in Theorem1.1).Linear determinacy clearly is an ideal situation from the point view of application.
If f(K)<max{f(u):u∈[0,K]},such as toxin-determined functional response[12], then we onlyfind an estimate of the lower bound of wave speed(Part(b)of Theorem1.1). For this case,it is unclear whether there exists a traveling wave connecting E K and E∗if
2
d
βf(K)−μ
≤c<2
d
βmax{f(u):0≤u≤K}−f(u∗)
.
In the case of(4.1),an additional,biologically important and mathematically interesting question is whether c∗is identical to another important quantity:the asymptotic spreading speed.The numerical result done in[2]for Holling Type I functional response suggested that the answer to this question is positive.We shall consider to give a theoretical study of this problem as our next research task.
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