Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 14, 669 - 677
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/nade.2016.6974
Existence, Uniqueness Solution of a Modified
Predator-Prey Model
M. A. Al Qudah
Mathematical Science Department
Princess Nourah Bint Abdulrahman University
P.O. Box 84428, Riyadh 11671, Saudi Arabia
Copyright © 2016 M. A. Al Qudah. This article is distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Abstract
A modified predator-prey model which describes the interaction between the
Predator-Prey species and at the species itself is considered. The existence and
uniqueness of solutions of a modified predator-prey model in 𝐿𝑝,𝑞 space is proved
under certain conditions of 𝑝 and 𝑞. Some properties of the dynamically system
of such model are described such as its stability. Graphs of solutions displayed.
Applications of the results in mathematical biology are discussed.
2000 Mathematics Subject Classification: 92B99
Keywords: Predator-Prey model, existence, uniqueness, Stability
1. Introduction
The dynamics of population has been described using mathematical models
which have been very successfully for studying animals and human populations.
Lotka [12] initiated the predator-prey models and competing species relations.
Levin and Segel [11] studied some biological hypotheses concerning the origin of
Planktonic Patchiness model. Fife [10] considered reaction and diffusion systems
which are distributed in 3-dimentional space or on a surface rather than on the
line. Calderon [9] studied the diffusion and non-linear population theory.
Abualrub [1] discussed diffusion problems in mathematical biology. In addition,
Abualrub [2] studied diffusion in 2- dimensional spaces for which diffusion is
more realistic and applicable in life and he proved the existence and uniqueness of
long range diffusion reaction model on population dynamics. Abualrub [4] proved
670
M. A. Al Qudah
the existence and uniqueness of solutions of a diffusive predator-prey model. AlQudah and Abualrub [5] studied the existence and uniqueness of long range
diffusion involving flux for insect dispersal model, and they studied in [7] solitary
and traveling wave solutions with stability analysis also for insect dispersal
model. In this paper, we make another modification of the model of Abualrub [3]
which is a modification to planktonic patchiness model, - a kind of predator-prey
model-, for more details see [11]. Existence and uniqueness of solutions for the
modified model will be considered. A traveling wave solutions and stability have
been considered. This paper is organized as follows. In Section 2, a modified
Predator- Prey model is considered. The existence and uniqueness of its solutions
is given in Section 3. The stability analysis and the conclusion are given in
Section 4.
2. The predator- prey model
The model we are going to consider here is another modification to the
planktonic patchiness model, a kind of predator-prey models, which were
originally considered by Levin and Segel in 1976 [11]. We use the model in [4, 3,
6] with the assumption that the species specific diffusion coefficients be constant
to come up with the predator- prey model:
𝑢𝑡 − ∆𝑢 = 𝑎1 𝑢 + 𝑎2 𝑢2 − 𝑎3 𝑢𝑣,
(1)
𝑣𝑡 − ∆𝑣 = 𝑎4 𝑢𝑣 − 𝑎5 𝑣 2 .
(2)
Where 𝑢 = 𝑢(𝑥, 𝑦) is the prey density population, 𝑣 = 𝑣(𝑥, 𝑦)is the predator
density population,
𝜕2
𝜕2
𝑥 = (𝑥1 , 𝑥2 ), and ∆= 𝜕𝑥 2 + 𝜕𝑥 2 represents the diffusion (dispersal), we shall
1
2
assume that 𝑎2 ,…,𝑎5 are constants and 𝑎1 may assumed to be compact supported
and bounded function of 𝑥; that is ( 𝑎1 = 𝑎1 (𝑥); 𝑎1 (𝑥) = 0 if |𝑥| > 𝑁; where N; is
a constant ) not a constant this is assumed, because the birth (or death) rate may
depend on the environment, which is assumed to be bounded. Another reason for
assuming that 𝑎2 ,…,𝑎5 are constants, is due to the fact that the birth(or death) rate
depends on the interaction between the male and the female (sexual interaction as
in terms 𝑎2 𝑢2 and −𝑎5 𝑣 2 ) or on the binary interaction between the males and
females of the prey and the predator respectively ( as in the term 𝑢𝑣). In this paper
we assume that 𝑎1 is constant and we modify the model by assuming another kind
of interaction between the prey and the predator, that is two terms are added,
namely to get the following model :
𝑢𝑡 − 𝐷1 ∆𝑢 = 𝑎1 𝑢 − 𝑎2 𝑢2 − 𝑎3 𝑢𝑣 − 𝑎4 𝑢𝑣 2 ,
(3)
𝑣𝑡 − 𝐷2 ∆𝑣 = 𝑎5 𝑢𝑣 − 𝑎6 𝑣 − 𝑎7 𝑣 2 + 𝑎8 𝑢𝑣 2 .
(4)
2
2
The added terms −𝑎4 𝑢𝑣 , and 𝑎8 𝑢𝑣 mean that there exists an interaction
between the species itself such as (in the mating period so we can consider the
term 𝑣 2 is the existence of a male and a female together, this will lead two predators
Existence, uniqueness solution of a modified predator-prey model
671
to meet on a prey. As the predator has a strong rapacity and can take more than it
needs so it is better to consider the term 𝑢𝑣 2 ), and then interact the two species
together in the environmental. The constants 𝑎1 ,𝑎2 ,…,𝑎8 are positive and
𝐷1 , 𝐷2 are the diffusion coefficients which are small positive constants. Finally
assume that the initial data are in the same 𝐿𝑝 space for some 𝑝 > 1, The initial
values for Eqs. (3) and (4) will be given by
𝑢(𝑥, 0) = 𝑔(𝑥), 𝑣(𝑥, 0) = ℎ(𝑥) respectively, where both 𝑔(𝑥) and ℎ(𝑥) ∈
𝐿𝑝 (𝑅 2 ). In addition we will consider small values of time 𝑡, since we are looking
for local solution in the usual diffusion, but for large values of time one should
talk about long range diffusion as in [2, 5].
3. Usual diffusion with 𝒑 = 𝒒 in the 𝐿𝑝,𝑞 norms
To easily solve Eqs. (3) and (4) we shall make the terms 𝑎1 𝑢 and −𝑎6 𝑣
disappear from Eqs. (3) and (4); to do this let 𝑢(𝑥, 𝑡) = 𝑒 𝛼𝑡 𝑤(𝑥, 𝑡), and 𝑣(𝑥, 𝑡) =
𝑒 𝛽𝑡 𝑧(𝑥, 𝑡) where 𝛼 = 𝑎1 , 𝛽 = −𝑎6. Therefore Eqs. (3) and (4) together with the
initial data become as follows:
(5)
𝑤𝑡 − 𝐷1 ∆𝑤 = −𝑎2 𝑒 𝛼𝑡 𝑤 2 − 𝑎3 𝑒 𝛽𝑡 𝑤𝑧 − 𝑎4 𝑒 2𝛽𝑡 𝑤𝑧 2 ,
𝑥 ∈ 𝑅2,
𝑤(𝑥, 0) = 𝑔(𝑥),
(6)
𝑧𝑡 − 𝐷2 ∆𝑧 = 𝑎5 𝑒 𝛼𝑡 𝑤𝑧 − 𝑎7 𝑒 𝛽𝑡 𝑧 2 + 𝑎8 𝑒 (𝛼+𝛽)𝑡 𝑤𝑧 2 ,
(7)
𝑧(𝑥, 0) = ℎ(𝑥),
𝑥 ∈ 𝑅2,
(8)
𝜕
since we have the heat operator 𝜕𝑡 − ∆ in the left hand side of Eqs.(5) and (7),
therefore 𝑤 and 𝑧 can be obtained by solving the following integral equations:
𝑡
𝑤 = ∫ ∫ 𝐾1 (𝑥 − 𝑦, 𝑡 − 𝜏)[−𝑎2 𝑒 𝛼𝜏 𝑤 2 − 𝑎3 𝑒 𝛽𝜏 𝑤𝑧 − 𝑎4 𝑒 2𝛽𝜏 𝑤𝑧 2 ] 𝑑𝑦𝑑𝜏
0
𝑅2
+ ∫ 𝐾1 (𝑥 − 𝑦, 𝑡) 𝑔(𝑦)𝑑𝑦,
(9)
𝑅2
𝑡
𝑧 = ∫ ∫ 𝐾2 (𝑥 − 𝑦, 𝑡 − 𝜏)[𝑎5 𝑒 𝛼𝜏 𝑤𝑧 − 𝑎7 𝑒 𝛽𝜏 𝑧 2 + 𝑎8 𝑒 (𝛼+𝛽)𝜏 𝑤𝑧 2 ] 𝑑𝑦𝑑𝜏
0
𝑅2
+ ∫ 𝐾2 (𝑥 − 𝑦, 𝑡) ℎ(𝑦)𝑑𝑦,
(10)
𝑅2
where 𝐾1 , 𝐾1 are the fundamental solutions of the heat equation; thus:
𝐷
−|𝑥|2
𝐷
−|𝑥|2
1
1
2
𝐾1 (𝑥, 𝑡) = 2𝜋𝑡
𝑒 2𝑡 and 𝐾2 (𝑥, 𝑡) = 2𝜋𝑡
𝑒 2𝑡 , |𝑥| = (𝑥12 + 𝑥22 )2 , 𝑡 > 0.
Let
𝐷 = 𝑚𝑎𝑥{𝐷1 , 𝐷2 } where 𝐷 is a small positive constant. And 𝐾1 , 𝐾1 can be
estimated as in [2]; then
672
M. A. Al Qudah
|𝐾1,2 (𝑥, 𝑡)| ≤
𝐷
1
.
(11)
(|𝑥|+𝑡 2 )2
Using the symbol ⊛ to represent the convolution in space and time while the
symbol ∗ is to represent the convolution in space only; we can rewrite Eqs. (9)
and (10) in a simpler way as follows:
𝑤 = 𝐾1 ⊛ [−𝑎2 𝑒 𝛼𝜏 𝑤 2 − 𝑎3 𝑒 𝛽𝜏 𝑤𝑧 − 𝑎4 𝑒 2𝛽𝜏 𝑤𝑧 2 ] + 𝐾1
∗ 𝑔,
(12)
𝑧 = 𝐾2 ⊛ [𝑎5 𝑒 𝛼𝜏 𝑤𝑧 − 𝑎7 𝑒 𝛽𝜏 𝑧 2 + 𝑎8 𝑒 (𝛼+𝛽)𝜏 𝑤𝑧 2 ] + 𝐾2
∗ ℎ.
(13)
where 𝑤 and 𝑧 are weak solutions of Eqs. (1), (2), (3) and (4) respectively, which
implies that the integrals in Eqs. (9) and (10) exist in the Lebesgue sense.
6
6
4
Lemma 1 If 𝑤(𝑥, 𝑡), 𝑧(𝑥, 𝑡)𝜖𝐿(2−𝜀,2−𝜀) (𝑅2 × [0,1]); 𝑎𝑛𝑑 𝑔(𝑥), ℎ(𝑥)𝜖𝐿(2−𝜀) (𝑅 2 ),
𝑡ℎ𝑒𝑛 𝑓𝑜𝑟 𝜀 > 0,
‖𝑇(𝑤)‖ 6 6
,
2−𝜀 2−𝜀
≤ 𝐶 ′ 6 ‖𝑤‖26
′′
6 + 𝐶 6 ‖𝑤‖ 6 , 6 ‖𝑧‖ 6 , 6
2−𝜀 2−𝜀
2−𝜀 2−𝜀
2−𝜀
2−𝜀 2−𝜀
2−𝜀
′′′
2
‖𝑤‖
‖𝑧‖
‖𝑔‖
+𝐶 6
6 6
8 8 +𝐶 4
4 ,
,
,
2−𝜀 2−𝜀
2−𝜀 2−𝜀
2−𝜀
2−𝜀
2−𝜀
‖𝑇(𝑧)‖ 6 6 ≤ 𝐴′ 6 ‖𝑤‖ 6 6 ‖𝑧‖ 6 6 − 𝐴′′6 ‖𝑧‖2 8 8
,
,
,
,
2−𝜀 2−𝜀
2−𝜀 2−𝜀
2−𝜀 2−𝜀
2−𝜀 2−𝜀
2−𝜀
2−𝜀
′′′
2
+ 𝐴 6 ‖𝑤‖ 6 , 6 ‖𝑧‖ 8 , 8 + 𝐴 4 ‖ℎ‖ 4 .
2−𝜀 2−𝜀
2−𝜀 2−𝜀
2−𝜀
2−𝜀
2−𝜀
,
𝑎𝑛𝑑
Proof : Consider 𝑇(𝑤) and 𝑇(𝑤) to be the image of 𝑤 and 𝑧 respectively, such
that:
𝑇(𝑤) = 𝐾1 ⊛ [−𝑎2 𝑒 𝛼𝜏 𝑤 2 − 𝑎3 𝑒 𝛽𝜏 𝑤𝑧 − 𝑎4 𝑒 2𝛽𝜏 𝑤𝑧 2 ] + 𝐾1
∗ 𝑔,
(14)
(𝛼+𝛽)𝜏
𝛼𝜏
𝛽𝜏 2
2
𝑇(𝑧) = 𝐾2 ⊛ [𝑎5 𝑒 𝑤𝑧 − 𝑎7 𝑒 𝑧 + 𝑎8 𝑒
𝑤𝑧 ] + 𝐾2
∗ ℎ.
(15)
Assume that 𝑡 takes small values in order to show the existence and uniqueness
of local solutions to Eqs. (14) and (15). Take the first, second, third, and fourth
terms on the right hand side of Eq. (15) we shall use exponents 𝑟, 𝑠, 𝑝, 𝑞
respectively, when considering the 𝐿𝑝 norm and we have the same argument for
Eq. (14). It is obvious from Eq. (11) that:
𝐷
0 ≤ |𝐾2 (𝑥, 𝑡)| ≤
;
(16)
1 2
(|𝑥| + 𝑡 2 )
using Eq. (16) and the same estimation of 𝐾2 as in [7] which is given by:
𝜀
|𝐾2 (𝑥, 𝑡)| ≤
𝐷𝑇 ′ 2
|𝑥|2−𝜃 𝑡1−
2−(𝜃+𝜀)
2
(17)
Existence, uniqueness solution of a modified predator-prey model
673
where 0 ≤ 𝑡 ≤ 𝑇 ′ . 𝑇′ is very small,0 < 𝜃 < 2 and the very small positive
constant 𝜀 is chosen such that 0 < 𝜃 + 𝜀 < 2. From Eqs. (14) and (17), we
obtain:
|𝑇(𝑧(𝑥, 𝑡))| ≤
𝑇′
𝜀
′
𝐷𝑎5 𝑒 𝛼𝑇 𝑇 ′ 2 ∫
−
|𝑤(𝑦, 𝜏)||𝑧(𝑦, 𝜏)|𝑑𝑦𝑑𝜏
∫
0 𝑅2
𝑇′
𝜀
′
𝐷𝑎7 𝑒 𝛼𝑇 𝑇 ′ 2 ∫
|𝑥 − 𝑦|2−𝜃 |𝑡 − 𝜏|1−
′
𝜀
𝑇′
+𝐷𝑎8 𝑒 (𝛼+𝛽)𝑇 𝑇 ′ 2 ∫0 ∫𝑅2
|𝑥 − 𝑦|2−𝜃 |𝑡 − 𝜏|1−
|𝑤(𝑦,𝜏)||𝑧(𝑦,𝜏)|2 𝑑𝑦𝑑𝜏
2−(𝜃+𝜀)
2
|𝑥−𝑦|2−𝜃 |𝑡−𝜏|1−
ℎ|.
𝑝
|𝑧(𝑦, 𝜏)|2 𝑑𝑦𝑑𝜏
∫
0 𝑅2
2
2
2−(𝜃+𝜀)
2
𝑞
2
2
2−(𝜃+𝜀)
2
+ |𝐾2 ∗
𝑞
2
2
2
𝑞
2
(18)
2
Now, assume that ℎ𝜖𝐿 (𝑅 ), 𝑧 𝜖𝐿 (𝑅 ), 𝑤 𝑧 𝜖𝐿 (𝑅 ), 𝑤𝑧 𝜖𝐿 (𝑅 ). Let 𝑟 >
1 be chosen such that:
1 2 𝜃
4
= − ; 2<𝑝< ,
(19)
𝑟 𝑝 2
𝜃
by taking the 𝐿𝑟 (𝑅 2 ) norm of the both sides and use the Benedek-Panzone
Potential Theorem, see [8], the first, the second and the third terms of the right
hand side of Eq. (18) become:
‖𝑇(𝑧(. , 𝑡))‖𝑟
′
𝜀 𝑇 ‖𝑤(. , 𝜏)‖𝑝 ‖𝑧(. , 𝜏)‖𝑝 𝑑𝜏
′2
≤ 𝐷𝑎5 𝑑𝑝 𝑇 ∫
2−(𝜃+𝜀)
0
|𝑡 − 𝜏|1− 2
′
′
2
2
𝜀 𝑇 ‖𝑧(. , 𝜏)‖𝑝 𝑑𝜏
𝜀 𝑇 ‖𝑤(. , 𝜏)‖𝑝 ‖𝑧(. , 𝜏)‖𝑝 𝑑𝜏
′2
′2
− 𝐷𝑎7 𝑑𝑝 𝑇 ∫
+
𝐷𝑎
𝑑
𝑇
∫
8 𝑝
2−(𝜃+𝜀)
2−(𝜃+𝜀)
0 |𝑡 − 𝜏|1−
0
2
|𝑡 − 𝜏|1− 2
+ ‖𝐾2 ∗ ℎ(. , 𝑡)‖𝑟 .
(20)
Let
𝑞
𝑞
𝑞
‖𝑤(. , 𝜏)‖𝑝 ‖𝑧(. , 𝜏)‖𝑝 𝜖𝐿2 (𝑅 + ), ‖𝑧(. , 𝜏)‖𝑝 2 𝜖𝐿2 (𝑅 + ), 𝑎𝑛𝑑 ‖𝑤(. , 𝜏)‖𝑝 ‖𝑧(. , 𝜏)‖𝑝 2 𝜖𝐿2 (𝑅 + ),
and let𝑠 > 1 such that for the third term we have:
1 3 2 − (𝜃 + 𝜀)
6
= −
; 3<𝑞<
,
(21)
𝑠 𝑞
2
2 − (𝜃 + 𝜀)
Again, by applying the Benedek-Panzone Potential Theorem, the first, the second
and the third terms of the right hand side of Eq. (20), and taking the 𝐿𝑠 (𝑅 + )norm
of the both sides, we obtain:
‖𝑇(𝑧)‖𝑟,𝑠 ≤ 𝐷𝑎5 𝑑𝑝 𝑑𝑞 ‖𝑤‖𝑟,𝑠 ‖𝑧‖𝑝,𝑞 − 𝐷𝑎7 𝑑𝑝 𝑑𝑞 ‖𝑧‖2𝑝,𝑞
+ 𝐷𝑎8 𝑑𝑝 𝑑𝑞 ‖𝑤‖𝑝,𝑞 ‖𝑧‖2𝑝,𝑞 + ‖𝐾2 ∗ ℎ‖𝑟 .
(22)
2
Now, take 𝑝 = 𝑟, then from Eq.(19) we get 𝑝 = 𝜃 𝑎𝑛𝑑 𝑞 = 𝑠, then from Eq.(21)
4
we have 𝑞 = 2−(𝜃+𝜀), 0 < 𝜃 < 2, 𝜀 > 0. In addition we shall require that 𝑝 = 𝑞.
674
M. A. Al Qudah
Therefor 𝜃 =
𝐷𝑎5 𝑑𝑝 𝑑𝑞 , 𝐴′′
𝐴
4
2−𝜀
‖ℎ‖
4
2−𝜀
2−𝜀
3
6
2−𝜀
; which implies that: = 𝑞 = 𝑟 = 𝑠 =
= 𝐷𝑎7 𝑑𝑝 𝑑𝑞 , and 𝐴′′′
6
2−𝜀
6
2−𝜀
, and selecting 𝐴′
= 𝐷𝑎8 𝑑𝑝 𝑑𝑞 . And since ‖𝐾2 ∗ ℎ‖
6
2−𝜀
6
2−𝜀
=
≤
follows directly from imbedding Lemma (namely Lemma 2) for the
initial data, this turns will conclude the main Theorem.
Lemma 2 Let 𝐹(𝑥, 𝑡) = 𝐾2 ∗ ℎ. Assume that ℎ ∈ 𝐿𝑟 (𝑅 2 ), 1 < 𝑟 < ∞. If for 𝑝, 𝑞 >
1
1
1
1 with 𝑝 + 2𝑞 = 𝑟 , then 𝐹(𝑥, 𝑡) ∈ 𝐿𝑝,𝑞 (𝑅 2 × 𝑅 + ) and ‖𝐹‖𝑝,𝑞 ≤ 𝑐(𝑟)‖ℎ‖𝑟 ,
where 𝑐(𝑟) is a constant depending on 𝑟 and the dimension.
Proof see [4].
4
Theorem 3 Suppose that the initial data 𝑔, ℎ ∈ 𝐿2−𝜀 (𝑅 2 ), 1 < 𝑟 < ∞, 0 < 𝜃 < 2,
if 𝜀 > 0 is very small such that ‖𝑔‖ 4 ‖ℎ‖ 4 < 𝜀, then ∃a unique solution
2−𝜀
𝑢, 𝑣 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ‖𝑢‖
6
6
,
2−𝜀 2−𝜀
, ‖𝑣‖
6
6
,
2−𝜀 2−𝜀
2−𝜀
< ∞; where 0 < 𝜀 < 1.
The proof of this theorem can be done using the same argument as the proof of
Lemma 1.
4. Stability Analysis
In this section we are discuss the stability of the modified model given by Eqs.
(3), (4) in one dimension. First we dimensionless the system by setting
1
𝑎2
𝑎3
1
1
𝑈 = 𝑎 𝑢, 𝑉 = 𝑎 𝑣, 𝑇 = 𝑎1 𝑡, 𝑋 =
𝑎32
𝑎1
, 𝑎7 = 𝑎3 𝑘1 , 𝑎8 =
𝑎3 𝑎5
𝑎1
𝑎 2
(𝐷1 ) 𝑥, 𝐷
2
𝐷
𝑎
𝑎6
= 𝐷1 , 𝑘1 = 𝑎5 , 𝑘2 = 𝑎
2
2
1 𝑘1
, 𝑎4 =
, then we have:
2
𝑈𝑇 = 𝐷𝑈𝑋𝑋 + 𝑈 − 𝑈 − 𝑈𝑉 − 𝑈𝑉 2 ,
𝑉𝑇 = 𝑘1 𝑉[𝑈 − 𝑘2 − 𝑉 + 𝑈𝑉] + 𝑉𝑋𝑋 .
(23)
It is clear that 𝑘1 , 𝑘2 are positive constants since 𝑎1 , 𝑎2 , … , 𝑎8 are positive using
the fact that only the predator are capable of moving toward the prey we have
𝐷1 = 0 and letting 𝑈(𝑋, 𝑇) = 𝑢1 (𝜉) and 𝑉(𝑋, 𝑇) = 𝑣1 (𝜉): 𝜉 = 𝑋 − 𝑐𝑇 where 𝑐 is
the wave speed which must be determined , thus system (23) become:
−𝑐
−𝑐
𝑑𝑢1
= 𝑢1 [1 − 𝑢1 − 𝑣1 − 𝑣1 2 ],
𝑑𝜉
𝑑𝑣1
𝑑 2 𝑣1
= 𝑘1 𝑣1 [𝑢1 − 𝑘2 − 𝑣1 − 𝑢1 𝑣1 ] +
,
𝑑𝜉
𝑑𝜉 2
(24)
𝑑𝑣
Using the method of reduction of order by setting m= 𝑑𝜉1 then we have a system of
nonlinear of first order ODE's:
Existence, uniqueness solution of a modified predator-prey model
675
𝑑𝑢1 −𝑢1
[1 − 𝑢1 − 𝑣1 − 𝑣1 2 ],
=
𝑑𝜉
𝑐
𝑑𝑣1
= 𝑚,
𝑑𝜉
𝑑𝑚
= −𝑐𝑚 − 𝑘1 𝑣1 [𝑢1 − 𝑘2 − 𝑣1 − 𝑢1 𝑣1 ].
𝑑𝜉
(25)
It is clear that system (25) is an almost linear system and has equilibrium points of
𝑑𝑢
𝑑𝑣
𝑑𝑚
the form (𝑢1 , 𝑣1 , 𝑚) by setting 𝑑𝜉1 = 𝑑𝜉1 = 𝑑𝜉 = 0 the equilibrium points
are:(0,0,0), (1,0,0) 𝑎𝑛𝑑 (0, −𝑘2 , 0). The Jacobian matrix corresponding system
(25) is:
𝐽=[
−[1−2𝑢1 −𝑣1 −𝑣1 2 ]
[𝑢1 +2𝑢1 𝑣1 ]
𝑐
𝑐
0
−𝑘1 𝑣1 + 𝑘1 𝑣1 2
0
−𝑘1 𝑢1 + 𝑘1 𝑘2 + 2𝑘1 𝑣1 − 2𝑘1 𝑢1 𝑣1
0
1 ].
−𝑐
It is clear that the points (0,0,0) 𝑎𝑛𝑑 (1,0,0) are unstable saddle points since the
corresponding eigenvalues of the characteristic equation for each equilibria are of
opposite signs, but the point (0, −𝑘2 , 0) is asymptotically stable because its
𝑘2 2 −𝑘2 −1
characteristic equation: (
roots: 𝜆1 =
𝜆2,3 =
𝑘2 2 −𝑘2 −1
𝑐
−𝑐±√𝑐 2 −4𝑘1 𝑘2
2
𝑐
− 𝜆) (𝜆2 + 𝑐𝜆 + 𝑘1 𝑘2 ) = 0 has the following
< 0 if 𝑘2 𝜖 [0,
1+√5
2
] since 𝑘2 𝑎𝑛𝑑 𝑐 are positive constants.
< 0 if 𝑐 > 2√𝑘1 𝑘2 then the point is node, and if
𝑐 < 2√𝑘1 𝑘2 then the point is spiral in each case the point is asymptotically
stable.
Conclusion
A- The wave speed 𝑐 depends on the 𝑎1 , 𝑎2 , 𝑎5 and 𝑎6 actually on 𝑎4 and 𝑎8
𝑎
(since 𝑎5 = 𝑎8 𝑎3 ) which are related to the coefficients of 𝑢𝑣 and 𝑢𝑣 2 . Thus the
4
coefficients of 𝑢𝑣 2 in the prey equation is less than the coefficient of 𝑢𝑣 , which
means that, the decay of the prey decreases after we added the term 𝑢𝑣 2 . Also in
the predator equation we have added the term 𝑢𝑣 2 with rate 𝑎8 , and 𝑎8 > 𝑎5,
which means that, the growth rate of the predator increases after we add the term
𝑢𝑣 2 . Therefore the interaction between the predator and the prey with the term
𝑢𝑣 2 is better than the interaction with the term 𝑢𝑣.
B- The interaction between two predators and one prey is possible even though
we are at closed environment that is for several reasons:
)i) The temperature has an effect on movement of the predator and prey. For
example if it increases, the predator's movement will slow down. And if it
decreases the prey will hide. Consequently, two predators will gather on a prey.
676
M. A. Al Qudah
(ii) In the mating period we can consider the term 𝑣 2 is the existence of a male
and a female together, this will lead two predators to meet on a prey. As the
predator has a strong rapacity and can take more than his needs so it is better to
consider the term 𝑢𝑣 2 .
In both cases, low temperature and mating, the consumption of food will
decrease and providing food will be continuous to all the species.
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Received: September 28, 2016; Published: November 15, 2016