e c o l o g i c a l m o d e l l i n g 2 1 6 ( 2 0 0 8 ) 415–416
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Letter to the Editor
Stability condition for the predator-free equilibrium point of
predator–prey systems with a class of functional response夽
Stability analysis of a predator–prey system incorporating a
given functional response is a popular issue in mathematical ecology (Hogarth et al., 1992; Chattopadhyay et al., 2002;
González-Olivares and Ramos-Jiliberto, 2003; Aghajani and
Moradifam, 2006; Canan, 2008; Moghadas and Corbett, 2008;
Wang, 2008). Hesaaraki and Moghadas (2001) have considered
predator–prey systems with a class of functional response.
They have obtained a sufficient and necessary condition for
the existence of limit cycles around the interior equilibrium
point of the considered system. They argue that “it is easy to
check that (0,0) and (1,0) are both saddle points”. However, I
find that the predator-free equilibrium point (1,0) can change
from the locally unstable state to be locally asymptotically
stable. Hence, it is partial to conclude that the equilibrium
point (1,0) is a saddle point. Moreover, the existence of limit
cycles around the equilibrium point E∗ = (x∗ , y∗ ) can be considered if and only if the predator-free equilibrium point is
unstable. Because if it is locally asymptotically stable, the equilibrium point (x∗ , y∗ ) lies in the fourth quadrant. It has no
ecological meanings. Therefore, we represent the following
proposition
Proposition 1. Assuming that (A1 ) − (A5 ) (see in Hesaaraki and
Moghadas, 2001) hold, we can obtain that
(1) If ϕ(1) > (D/), the equilibrium point (1,0) is a saddle point, and
hence, is locally unstable.
(2) If ϕ(1) < (D/), the equilibrium point (1,0) is locally asymptotically stable.
Proof. The Jacobia matrix of the system (1.1) (see in Hesaaraki
and Moghadas, 2001) at the equilibrium point (1,0) is J1 =
夽
−r
−ϕ(1)
0
ϕ(1) − D
It is easy to show that the two eigenvalues are 1 = −r and
2 = ϕ(1) − D.
Clearly, the first one is negative.
Hence, the stability of the equilibrium point (1,0) depends
only on the sign of the second eigenvalue 2 = ϕ(1) − D.
If 2 = ϕ(1) − D > 0, i.e. ϕ(1) > (D/), the equilibrium point
(1,0) is a saddle point and locally unstable.
According to the assumptions A1 and A2 , the response
function ϕ(x) is increasing and positive. Thus we have
ϕ(1) > 0.
From the assumption A4 , the constant c is the maximum value
of the response function ϕ(x). That is to say
ϕ(1) < c.
Hence, we obtain that
0 < ϕ(x∗ ) =
D
< ϕ(1) < c.
Thus, x∗ < 1. The equilibrium point E∗ = (x∗ , y∗ ) lies in the first
quadrant and is a positive point.
Again, if 2 = ϕ(1) − D < 0, i.e. ϕ(1) < (D/), the equilibrium
point (1,0) is locally asymptotically stable.
Similarly, we have
0 < ϕ(1) <
D
= ϕ(x∗ ) < c
Hence, x∗ > 1. The equilibrium point E∗ = (x∗ , y∗ ) lies in the
forth quadrant.
If the equilibrium point E∗ = (x∗ , y∗ ) is not a positive point,
we do not think it has ecological meanings. Thus, we cannot
consider its stability property and the existence of limit cycles
around it.
Therefore, the theorem 1.1 (see in Hesaaraki and
Moghadas, 2001) should be proved if and only if the predatorfree equilibrium point is unstable.
Supported by the National Natural Science Foundation of China (No. 30700100).
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e c o l o g i c a l m o d e l l i n g 2 1 6 ( 2 0 0 8 ) 415–416
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Zhi-hui Ma ∗
Zi-zhen Li
Key Laboratory of Arid and Grassland Agroecology of Ministry of
Education, Lanzhou 730000, China
∗ Corresponding author. Tel.: +86 931 8913370;
fax: +86 931 8912823.
E-mail addresses: [email protected] (Z.-h. Ma),
[email protected] (Z.-z. Li)
27 March 2008
Published on line 26 June 2008
0304-3800/$ – see front matter
© 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2008.05.002