Existence of non-constant positive stationary solutions of the shadow predator-prey systems with Allee effect
School of Mathematics, Liaoning Normal University
School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology
In this paper, we consider the dynamics of the shadow system of a kind of homogeneous diffusive predator-prey system with a strong Allee effect in prey. We mainly use the time-mapping methods to prove the existence and non-existence of the non-constant positive stationary solutions of the system in the one dimensional spatial domain. The problem is assumed to be subject to homogeneous Neumann boundary conditions. MSC: 35K57; 35B09
ux(x, t) = ux(x, t) = vx(x, t) = vx(x, t) = , t > , x = , h ,
u(x, ) = u(x) , v(x, ) = v(x) , x (, h ).
Here u = u(x, t) and v = v(x, t) stand for the densities of the prey and predator at time t >
and a spatial position x (, h ) with h (, ), respectively; d, d > are the
diffusion coefficients of the species; d is the death rate of the predator, a measures the
saturation effect, m is the strength of the interaction. The Allee threshold b is assumed to be
smaller than . The strong Allee effect introduces a population threshold, and the
population must surpass this threshold to grow. The boundary condition here is assumed to be
homogeneous Neumann type, which implies that there is no flux for the populations on
the boundary. For more details on the problem (.), we refer interested readers to 
and references therein.
In , the authors considered the traveling wave solutions of system (.). More precisely,
they showed that there is a non-negative traveling wave solution of system (.) connecting
the semi-trivial solution (b, ) and the positive equilibrium solution (u, v). They also
proved that, under certain suitable conditions, there is a small traveling wave train solution
of system (.).
In , the authors considered the non-existence of non-constant positive steady state
solutions, and bifurcations of spatially homogeneous and non-homogeneous periodic
solutions as well as non-constant steady state solutions are studied. These results allow for
the phenomenon that the rich impact of the Allee effect essentially increases the system
Although the existence and non-existence of non-constant steady state solutions of the
system (.) has been considered in  for finite diffusion coefficients, no results have been
reported to consider the existence and non-existence of the positive non-constant steady
state solutions for the shadow system corresponding to the system (.). The shadow
system we mentioned here stands for the system where one of the diffusion coefficients tends
to infinity. The readers are referred to  for the earlier contributions on the shadow
Thus, the purpose of this paper is to consider the existence and non-existence of positive
non-constant solutions of the following elliptic equations:
ux(x) = ux(x) = vx(x) = vx(x) = , x = , h ,
where d .
The methods we used in the paper are standard time-mapping methods (see  and
references therein for precise details on time-mapping methods). We hope that the results
in the paper will allow for a clearer understanding of the rich dynamics of this particular
pattern formation system. In Section , we state the derivation of the shadow system of
the original reaction-diffusion system (.). In Section , we study the existence of the
non-constant stationary solutions of the shadow system; in Section , we end up our
discussions by drawing some conclusions.
2 Derivation of the shadow system
Firstly, we state the following useful a priori estimate for the non-negative solutions of
system (.) obtained in :
Lemma . Suppose that d, d, a, b, d, m, h > , and that (u(x), v(x)) is a non-negative
steady state solution of (.). Then either (u, v) is one of constant solutions: (, ), or (b, ),
or for x [, h ], (u(x), v(x)) satisfies
< u(x) < and
am+uvu dv = .
For later use in our discussion, we rewrite the second equation of system (.) in the
duxx = u( u)( ub ) ma+uu ,
subject to the additional condition (.) and the condition
Thus, system (.) reduces to the following single -parameterized scalar
Lemma . implies that the non-negative solutions of the system (.) is bounded. Then
we know that vxx as d . Since our problem is of Neumann boundary condition
type, v is a constant, say . As d , there exists a positive number C = C(d) > , such
3 Existence of non-constant positive stationary solutions of the shadow system
In this section, we mainly concentrate on the existence of the non-constant positive
solutions of the reduced shadow system (.).
For the purpose of our investigations, we define
f (u) = u( u) ub am+uu
Then we introduce the following energy functional:
From (.), we can find that, for any x (, h ), E (x) , and F(u(x)) < F() = F( ),
where := u(), and := u(h ).
It follows that if u = u(x) is a solution of (.), then F(u) must attain its local minimal
value at a point in (, ).
We rewrite f (u) as
(u ) := (a + u)(kbu)(u b) .
We have the following lemma on the properties of the function (u ) defined above.
:= b + a + (b + a) + (ab + a b) ,
:= b + a (b + a) + (ab + a b) .
Proof It is obvious that for any u (, b) (, ), we have (u ) < , while (u ) > for any
u (b, ). We can directly check that
with b < u() < < u+() < .
Since f (u) < for < u < u(), f (u) > for u() < u < u+(), it follows that F(u) is
convex in (, u+()), and concave in (u(), ), and F(u) taking its local minimum value
at u = u(). In other words, F(u) is decreasing in (, u()) (u+(), ), and increasing
in (u(), u+()).
Because > > provided that ab + a b < , we conclude from (.) that (u ) is
increasing ( (u) > ) for u (, ), while for any u (, ) (, ), (u ) is decreasing
( (u) < ). The second part of the lemma can be proved similarly.
From Lemma ., it follows that (u ) attains its maximum value := ( ) at u = . If
> holds, then f (u) < for all u (, ). Thus, F(u) does not has its minimal value
point in (, ), which implies that the shadow system (.) does not possess positive
nonconstant stationary solutions. Similarly, if = holds, then system (.) does not also
possess positive non-constant stationary solutions.
Thus, in order for the shadow system to have non-constant positive stationary solutions,
we need to concentrate on the case when (, ).
In this case, there exist two zeros of f (u) = , and we can denote them by
Thus, the problem admits solutions for some h > if and only if (, ) and we are
now deriving the precise information on the suitable h > such that the problem has
positive non-constant stationary solutions.
If F() F(u+()) holds, then there exists a unique (, u()), such that F() =
Then for any (, u()), there exists a unique (u(), u+()), such that F() =
By the definition of E(x) = d(ux(x))/ + F(u(x)), and the fact that E (x) , we have
Then s = g(u) is well defined and is strictly increasing in (, ), since in this interval
F(u) is convex and takes a strict minimum at u = u().
Let p > be given by
g (p cos t) dt,
:= , if F() F(u+()),
, if F() F(u+()).
In the following, we want to show the limit of h() as u(). In fact, following the
same argument in  (say, for example, pp.-), one can verify that
By the same argument as on p. of , we have
g (s) =
g (s) =
g (s) = fg((us)) H(u),
H(u) := f (u)f (u)F(u) + f (u) f (u) f (u)F(u) .
By (.), we have
f (u) = u( u) ub m a +u u .
Then after direct calculations, we have
h (p) =
cos tg (s) dt,
h (p) =
s = p cos t.
h () = dhd(p) = g ()
cos t dt = ,
and h (p) > due to g (s) > , it follows that h (p) > or equivalently dh/dp > . This
together with the fact that dp/d < , we can conclude that dh/d < .
Summarizing the analysis above, we can conclude the following.
the shadow system (.) with the condition (.) has no non-constant positive stationary
h > h :=
In this paper, we studied the existence and non-existence of the positive non-constant
stationary solutions of a shadow system corresponding to a kind of diffusive homogeneous
predator-prey system with Holling type-II functional response and strong Allee effect in
prey. We hope that the results in the paper will allow for the clearer understanding of
the rich dynamics of this particular pattern formation system. Future work might include
considering the qualitative behavior of the parabolic shadow system.
The authors declare that they have no competing interests.
All authors have equal contributions. All authors read and approved the final manuscript.
The authors are very grateful to the anonymous referee for his/her valuable comments and suggestions, which led to an
improved presentation of the manuscript.