On the Free Boundary Problem for the Predator-Prey Model
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On the Free Boundary Problem for the Predator-Prey Model Journal of Physics: Conference Series A.
Institute
of Mathematics,
Tashkent, Uzbekistan
100174,
Tashkent
city,
Olmazor
district,
University 46th
Street,
E-mail:
elmurodov8111@mail.ru, Abstract. In this article, we investigate the free boundary problem for the classical predator- prey model with double free boundaries. This system mimics the spread of invasive or new predator species, in which free boundaries represent the expanding fronts of predator species and are described by the Stefan condition. For this system, the existence and uniqueness of solutions is checked, and the behavior of positive solutions is also considered. A priori estimates for the required functions are established. For this model, the dichotomy of spread and disappearance has been proven. Free boundary; Biological invasions; Predator-prey; Spreading-vanishing dichotomy. 1. Introduction Migration
of
a
new or
invasive
species is
one
of the
most
important
topics
in
mathematical
ecology. Many
mathematicians
have
tried
to
develop
various invasion
models
and
investigate
them
from
the
point
of
view
of mathematical
ecology.
For
example,
in
[1-2],
population models
with
a
free
reaction-diffusion boundary
are
proposed
in
order
to understand
the
process
of
creating
a new
or
invasive
population. In
fact,
the spatial
distribution
of
prey
and predator
is
not
uniform within
a
fixed
bounded areal,
so
it
is more
realistic
to
introduce
reaction-diffusion equations
to
describe
the spatial
distribution
of
each
species. H.W.
Yin
et
al.
Investigated a
modified
Leslie-Gower predator-prey
model
with
Crowley-Martin
functional response
and
spatial
diffusion
under
uniform
Neumann
boundary
conditions
[3].
They
obtained
the
existence
of
a
global positive
solution,
as
well
as local
and
global
asymptotic
stability of
constant
equilibria. In
addition,
they found
the
presence
and
absence
of
intermittent
positive stable
states.
Our main goal is to investigate the long-term behavior of a predator-prey model with a Leslie-Gower free boundary. In this article, we consider the following model: To find functions u ( t, x ), v ( t, x
), s ( t ), h ( t ) in the domain ( D ⊂ Q ) Q = { ( t, x ) : 0 < t ≤ T, − L < x < L } , D
= { ( t, x ) : 0
< t ≤ T, h ( t ) < x < s ( t ) } satisfying the conditions u t − u xx − c 1 u x = u (1 − u ) − v u u + m , ( t, x ) ∈ Q, (1)
v t − dv xx − c 2 v x = kv 1 − bv u + a , ( t, x ) ∈ D, (2) u (0 , x ) = u 0 ( x ) , − L ≤ x ≤ L, (3)
v (0 , x ) = v 0 ( x ) , − s 0 ≤ x ≤ s 0 , (4)
u x ( t, − L ) = u ( t, L ) = 0
, t >
0 , (5) v ( t, h ( t )) = v ( t, s ( t )) = 0 t > 0 , (6) ˙ s ( t ) = − µv x ( t, s
( t )) , t >
0 , (7) ˙ h ( t ) =
− µv x ( t, h
( t )) , t >
0 . (8) where - free (unknown) boundaries h ( t ) , s ( t ) , which represent the front of propagation, is determined together with the functions u ( t, x ), v ( t, x
); d − diffusion coefficient, s 0 , L , µ , m , c i ( i = 1 , 2),
a and
b - are positive constants. The initial data ( u 0 , v 0 ) satisfy: ◦ i. u
0 ( x ) ∈ C 2+ α [ − L, L
] , u
0 ( − L ) = 0
, u 0 ( L ) = 0; 0
< u 0 ( x )
1 ,
L ≤ x ≤ L ; ◦ ii. v
0 ( x ) ∈ C 2+ α [ − s 0 , s 0 ] , v 0 ( − s 0 ) = 0 , v
0 ( s 0 ) = 0; 0
< v 0 ( x )
2 ,
s 0 ≤ x ≤ s 0 . Here and in what follows, M will denote constants depending on the data of the problem. A similar problem was studied in [4]. 2. A priori estimates In this section, we establish some a priori estimates of Schauder type that will be used to prove the global solvability of the problem. At the same time, the principles of maximum and comparison theorems are widely accepted. There are various methods for obtaining a priori estimates. In this paper, we will apply the method of obtaining a priori estimates proposed by S.N. Kruzhkov [5]. Therefore, we will adhere to the notation adopted in [5]. Lemma 1. Let
u ( t, x ) , v
( t, x
) , s
( t ) be a solution to problem (1) - (8). Then 0 < u ( t, x ) ≤ M 1 , t > 0 , ¯ Q, 0
( t, x
) ≤ M 2 , t > 0 , ¯ D, 0
˙ s
t ) ≤ M 3 , 0 < ˙ h ( t ) ≤ M 4 , t >
0 , where M 3 , M 4 are constants depending on µ, d, k, M 1 , M 2 . We will establish H¨ older norm bounds | · | 1+
and | · |
2+ α in ¯ D and ¯
Q . For each equation of the system, we formulate the corresponding problem u xx + b 1 ( u, v, u x ) − u t = 0 , Q, u (0 , x ) = u 0 ( x ) , − L ≤ x ≤ L, u x ( t, − L ) =
u ( t, L ) = 0 , 0 ≤ t ≤ T, (9)
dv xx + b 2 ( u, v, v
x ) − v t = 0 , D, v (0 , x
) = v 0 ( x ) , − s 0 ≤ x ≤ s 0 , v ( t, h ( t )) = v ( t, s ( t )) = 0 , 0 ≤ t ≤ T, ˙ s ( t ) = − µv x ( t, s
( t )) , 0 ≤ t ≤ T, ˙ h ( t ) =
− µv x ( t, h
( t )) , 0 ≤ t ≤ T, (10) where
b 1 ( u, v, u x ) = c 1 u x + u (1 − u ) − v u u + m , b
2 ( u, v, v x ) =
c 2 v x + kv 1 − bv u + a . Theorem 2. Assume that u ( t, x ) , u x ( t, x ) are continuous in Q T and suppose that u ( x, t ) is a solution for the problem (9). Then | u x ( t, x
) | ≤
C 1 ( M 1 ) , ( t, x ) ∈ ¯ Q T . (11) Moreover, if the weak second derivatives u xx , u tx are in L 2 ( Q T ), then there exists α = α ( M 1 , s ), such that | u
1+ α,Q
T ≤ C 2 ( M 1 , C
1 ) . (12) Additionally, assume that, u (
) satisfying (9) in ¯ Q T , is continuous with its derivatives u t , u x , u xx and
| u | 2+ α, ¯ Q T
∞ .
| u | 2+ α, ¯ Q T ≤ C 3 ( M 1 , C 1 , C
2 ) . (13) Proof:
The estimates (11)-(13) for ( t, x
) ∈ Q are immediate consequences of the results of [5].
In the case of problem (10), a priori estimates are constructed as follows. Estimates in the interior of the domain are established as in the case of problem (9). Further, replacing τ =
= 2 x s ( τ ) − h ( τ ) − s ( τ )+ h ( τ ) s ( τ ) − h ( τ ) , we straighten out the boundary. Then domain D T is mapped to domain Ω = { (
) : 0 < τ < T, − 1 < y < 1 } and for the function w ( τ, y ) =
v ( τ, u, s ( τ ) , h ( τ )), we obtain an equation with bounded coefficients and the right-hand side. By the results of [5], we establish estimates for | w y | , | w | 1+ γ up to the right boundary. Estimates for the highest derivatives are obtained from the results for linear equations [6]. Now let us prove that the free boundaries do not cross the lateral boundaries in the considered time interval. First, we get a new representation for the free boundary. Integrating (2) over D , we obtain t Z 0 dη s ( η ) Z 0 " dv ξ + 1 2 m 2 v 2 ξ − v η # dξ + k t Z 0 dη s ( η ) Z 0 v 1 − bv u + a dξ = 0 . We get
d µ ( s ( t ) − h ( t )) = s 0 Z − s 0 v 0 ( ξ ) dξ − s ( t ) Z h ( t ) v ( t, ξ ) dξ + k t Z 0 dη s ( η ) Z − h ( η ) v 1 − bv u + a dξ. (14) Theorem 3. Let u
t, x ), v ( t, x
), s ( t ), h ( t ) be a solution to (1)-(8). Then g ( t ) < L, where
g ( t ) = | s ( t ) − h ( t ) | > 0. Proof: We use relation (9): d µ g ( t ) + s ( t ) Z h ( t ) v ( t, ξ ) dξ − k t Z 0 dη s ( η ) Z h ( η ) v 1 − bv u + a dξ = s 0 Z − s 0 v 0 ( ξ ) dξ. Where d µ g ( t ) − k t Z 0 dη s ( η ) Z 0 v ( η, ξ ) dξ ≤ K, where K = s 0 R − s 0 v 0 ( ξ ) dξ . In the same way, taking into account the inequalities 0 ≤ s ( η ) R h ( η ) v ( η, ξ
) dξ ≤ M 2 ( s ( η ) − h ( η )) , d µ g ( t ) − kM 2 t Z 0 g ( η ) dη ≤ K,
we have g ( t ) ≤ m + n t Z 0 g ( η ) dη,
where m = µK d , n = kµM
2 d . Then g ( t ) =
me nt
0
≤ T = 1 n ln L m . Theorem 3 is proved. Lemma 4. (Comparison principle) Let
u ( t, x ), v ( t, x ), s ( t ), h ( t )) - be a solution to problem (1) - (8) with initial data ( u 0
x ), v 0 ( x )). a) Suppose that ( w ( t, x ), z 1 ( t ), δ 1 ( t )) satisfies
w t − dw xx − c 2 w x ≥ kw (1 − bw a ) , t > 0 , δ 1 ( t )
1 (
) , w ( t, z
1 ( t )) = 0 , w
( t, δ
1 ( t )) = 0 , t > 0 , ˙ z 1 ( t ) ≥ − µw x ( t, z 1 ( t )) , t > 0 . ˙ δ 1 ( t ) ≤ − µw x ( t, δ
1 ( t )) , t > 0 . If w (0 , x ) ≥ v 0 ( x ) in [ − L, L ] and z 1 (0) ≥ s (0) , δ 1 (0)
≤ h (0) , then
• z 1 ( t ) ≥ s ( t ) δ 1 ( t ) ≤ h ( t ) for t ≥ 0 , • w ( t, x ) ≥ v ( t, x
) for x ∈ [ h ( t ) , s ( t )] . b) Suppose that ( ϑ ( t, x ) , z
2 ( t ) , δ
2 ( t )) satisfies ϑ t − dϑ xx − c 2 ϑ x ≤ kϑ (1 − bϑ M 1 + a ) t > 0 δ 2 ( t ) < x < z 2 ( t ) , ϑ ( t, δ 2 ( t )) = 0 , ϑ
( t, z
2 ( t )) = 0 , t > 0 , z 2 ( t ) ≤ −
µϑ x ( t, z 2 ( t )) , δ 2 ( t ) ≥ − µϑ x ( t, δ 2 ( t )) , t > 0 . If ϑ (0 , x ) ≤ v 0 ( x ) in [ − L, L ], z 2 (0) ≤ s (0) , δ 2 (0)
≥ h (0) , then
• z 2 ( t ) ≤ s ( t ) δ 2 ( t ) ≥ h ( t ) for t ≥ 0. • ϑ ( t, x
) ≤ v ( t, x
) for x ∈ [ h ( t ) , s ( t )] . 3. Some qualitative properties of solutions Theorem 5. Let
u ( t, x ), v ( t, x ), s ( t ), h ( t ) be a solution to problem (1) - (8). If s ∞ > L , then lim t → + ∞ sup u ( t, x
) ≤ ¯ u ( x ); lim
t → + ∞ sup u
( t, x
) ≥ u ( x ); lim t → + ∞ sup v ( t, x
) ≤ ¯ v ( x ); lim
t → + ∞ sup v
( t, x
) ≥ v ( x ) . where ¯
u ( x ), u ( x ), ¯
v ( x ), v ( x ) upper and lower solutions to the problem. Theorem 6. Let
u ( t, x ), v ( t, x ), s ( t ), h ( t ) be a solution to problem (1) - (8). If s ∞ < L , then lim t → + ∞ sup u
( t, · ) ≥ ¯ u ( x ) for x ∈ [ − L, L ] and lim t → + ∞ sup k v ( t, · ) k C [ h ( t ) ,s ( t )] = 0 . 4. Uniqueness and existence of a solution Let’s use the representations for the unknown boundary (14). Theorem 7. Let conditions i. -
hold., Lemma 1 and Theorem 2. Then the solution to problem (1) - (8) is unique. Proof: We first establish the result for smaller values of t , and then extend the proof to the general case of 0
∞ . Assume that s 1 ( t ) , h 1 ( t ) , u 1 ( x, t ) , v
1 ( x, t ) and s 2 ( t ) , h 2 ( t ) , u 2 ( x, t ) , v
2 ( x, t ) are the solutions of the problem (14) and let y 1
t ) = max(
h 1 ( t ) , h 2 ( t )) , y 2 ( t ) = min( h 1 ( t ) , h 2 ( t )) , z 1 ( t ) = max(
s 1 ( t ) , s 2 ( t )) , z 2 ( t ) = min( s 1 ( t ) , s 2 ( t )) . Then each pair satisfies the identity (14). Subtracting, we obtain that d µ
g 1 ( t ) − g 2 ( t )) − ( h 1 ( t ) − h 2 ( t )) ≤ z 2 ( t ) Z y 1 ( t ) ( v 1 ( t, ξ
) − v 2 ( t, ξ )) dξ + y 1 ( t ) Z y 2 ( t ) v i ( ξ, t ) dξ + µ z 1 ( t ) Z z 2 ( t ) v i ( ξ, t ) dξ + + t Z 0 dη z 2 ( η ) Z y 1 ( η ) ( f ( u 1 , v
1 ) − f ( u 2 , v
2 )) dξ + t Z 0 dη z 1 ( η ) Z z 2 ( η ) f ( u i , v i ) dξ + t Z 0 dη y 1 ( η ) Z y 2 ( η ) f ( u i , v i ) dξ, (15) where
f ( u i , v
i ) =
v i 1 − bv i u i + a are the solution between y i
t ) and
z i ( t )( i = 1 , 2), i.., ( u i ( t, x
) , v
i ( t, x )) = ( u 1 ( t, x ) , v
1 ( t, x )) , h 2 , s
2 < h 1 , s 1 , ( u 2 ( t, x ) , v 2 ( t, x )) , h 2 , s
2 > h
1 , s
1 . From Lemma 1, we have that | v 1 ( t, y
1 ( t )) − v 2 ( t, y 1 ( t )) | ≤
M 5 | h 1 ( t ) − h 2 ( t ) | , | v 1 ( t, z 2 ( t )) − v 2 ( t, z 2 ( t )) | ≤
M 5 | s 1 ( t ) − s 2 ( t ) | , where
M 5 = max D | v x ( t, x ) | . Considering the difference U ( t, x ) =
u 1 ( t, x ) − u 2 ( t, x ), V ( t, x
) = v 1 ( t, x
) − v 2 ( t, x ), we obtain an equation with bounded coefficients and the problems
U t − U xx − c 1 U x − d 1 ( · ) U = F 1 ( · ) V, Q, U (0 , x ) = 0 , − L ≤ x ≤ L, U x ( t, − L ) = U ( t, L ) = 0 , 0 ≤ t ≤ T, (16)
V t − dV xx c 2 ( · ) V x − d 1 ( · ) V = F 2 ( · ) U, D, V (0 , x ) = 0
, − s 0 ≤ x ≤ s 0 , , V ( t, y 2 ( t )) ≤ M 7 max 0 ≤ η ≤ t | h 1 ( η ) − h 2 ( η ) | , 0 ≤ t ≤ T, V ( t, z 2 ( t )) ≤ M 7 max
0 ≤ η ≤ t | s 1 ( η ) − s 2 ( η ) | , 0 ≤ t ≤ T, (17) where
b i , c i , F i ( i = 1 , 2) limited and continuous functions. From the problem (16), (17), by the maximum principle, we find the estimates | U ( t, x
) | ≤
N 3 max Q | V ( t, x
) | , | V ( t, x ) | ≤ N 4 max 0 ≤ η ≤ t | s 1 ( η ) − s 2 ( η ) | + max 0 ≤ η ≤ t | h 1 ( η ) − h 2 ( η ) | + N 4 max
Q | U ( t, x
) | . By virtue of the established estimates for the functions u ( t, x ), v ( t, x
), s ( t ), h ( t ) can evaluate members from (15). Further, using the ideas and results of [7], the proof of the theorem is completed. Theorem 8. Let the conditions of Lemma 1 and Theorems 5 be satisfied. Then there exists a solution u ( x, t ) ∈ C 2+ γ ¯ D T , v ( x, t ) ∈ C 2+ γ ¯ Q T , s ( t ) ∈ C 1+ γ ([0 , T
]), h ( t ) ∈ C 1+ γ ([0 , T
]) of problem (1) - (8). To prove the existence of a solution to problem (1) - (8), we use the Leray Schauder theorem [6].
References [1] Wang, MX: The diffusive logistic equation with a free boundary and sign-changing coefficient. J. Differ. Equ. 258,1252-1266 (2015) [2] Du, Y, Matsuzawa, H, Zhou, M: Sprading speed and profile for nonlinear Stefan problems in high space dimensions.J. Math. Pures Appl. 103(2015), 741-787. [3] H.W. Yin, et al. Pattern analysis of a modified LeslieGower predatorprey model with CrowleyMartin functional response and diffusion, Comput. Math. Appl. 67 (8) (2014) 1607621. [4] Liu Y. et al. Biological invasion in a predatorprey model with a free boundary. Liu et al. Boundary Value Problems (2019) 2019:33 https://doi.org/10.1186/s13661-019-1147-7 [5] Kruzhkov S. N., Nonlinear parabolic equations with two independent variables // Transaction of the Moscow Mathematical Society. 1967, v.16,pp. 329 - 346 [6] Ladyzenskaya, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, AMS, Providence, RI (1988). [7] Takhirov J.O., Rasulov M.S. Problem with Free Boundary for Systems of Equations of Reaction-Diffusion Type, Ukrainian Math.J. (2018), 69(12), pp. 1968–1980. Download 244 Kb. Do'stlaringiz bilan baham:
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