Global Dynamics of A PDE Model For Eradication of Invasive Species

Volume 4, Issue 4, April – 2019 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
Global Dynamics of a PDE Model for Eradication of

Invasive Species
Rana D. Parshad Said Kouachi Jingjing Lyu
Department of Mathematics, Department of Mathematics, Department of Mathematics,
Iowa State University, College of Science, Qassim University, Clarkson University
Ames, IA 50011, USA. P.O.Box 6666, Buraydah 51452, KSA. Potsdam NY13699, USA.
Abstract:- The purpose of the current manuscript is to reversed females bearing two Y chromosomes, i.e.
propose a generic method that causes the local feminized supermales (r), at a constant rate µ to a target
extinction of a harmful invasive species. Eradication is population containing f and m. Mating between the
achieved via introduction of phenotypically modified introduced r and the wild-type m generates a
organisms into a target population. Here we propose a disproportionate number of males over time. The higher
model without the logistic type term, of which the incidence of males decrease the female to male ratio.
reaction terms may change sign, and so the solutions are Ultimately, the number of f decline to zero, causing local
not bounded a priori. We prove global existence of extinction. This theoretical method of eradication is known
solutions via a Lyapunov function method, and show as Trojan Y Chromosome strategy (TYC), [13]. Note, if an
existence of a finite dimensional (L2(Ω);H2(Ω)) global invasive species is used as a biological weapon, one would
attractor that supports states of extinction, improving aim at maximum damage, by choosing a species that might
current results in the literature. We also conduct populate very rapidly, and not grow according to the
numerical simulations to investigate the decay rate of logistic control terms (at least in certain time windows),
the female species. Lastly we apply optimal control assumed traditionally [51, 52]. There is a large literature of
techniques to compare the effectiveness of various such rapid population actuations in the so called case of an
reaction terms on species extinction. insect \outbreak" [4]. Furthermore, past models have not
considered the effects of directed movements, such as
Keywords:- Reaction Diffusion System, Global Existence, movement of the males and supermales towards high
Global Attractor, Optimal Control, Invasive Species, concentrations of females, or avoidance of high
Biological Control. concentrations, of each other. Thus such situations also
need to be considered in our setting.
I. INTRODUCTION
The TYC model has been intensely investigated
An exotic species commonly referred to as invasive recently [13, 42, 41, 43, 14, 66, 63, 59, 64, 44, 45, 61, 15],
species, is any species capable of propagating into a non- and in the case of the classical TYC model, we now know
native environment. As a result of globalization, exotic the attractor is actually in Hs, 8s _ 0, [66]. However, a
species are being introduced to ecosystems around the number of fundamental questions remain unanswered
world at an unprecedented pace, in many cases causing concerning existence of solutions as well as the existence
harm to the environment, human health, and/or the and regularity of a global attractor, in the case that the
economy [35, 34]. Once an exotic species is established in reaction terms are \bad", that is say without logistic control
a new environment, its detrimental potential might be terms, so that no a priori bounds on the solutions are
realized in the form of economic losses or threats to public possible. In [46] we began a program where we study TYC
health. Eradication initiatives in these cases frequently type models for biological control, where we remove the
require continuous efforts for long periods of time. A small logistic type term. We also assume nonlinear and
fraction of the estimated 50,000 exotic species in the US is functionally dependent birth and death rates, instead of the
harmful, but they inflict considerable damage [50, 16]. constant coefficient birth and death rates, assumed earlier.
Studies indicate losses of about $120 billion/year by 2004 In this case the system poses serious mathematical
[50]. A strategy for eradication of exotic species in which a difficulties, as the nonlinearities change sign, and the
\Trojan individual" is strategy is relevant to species with an components of the solution are not priori bounded in some
XY sex-determination system, in which males are the Lp space. There is extensive literature on such problems [1,
heterogametic sex (carrying one X chromosome and one Y 18, 20, 21, 25, 39, and 65]. In [46] we were able to use an
chromosome, XY) and females are the homogametic sex elegant Lyapunov functional to prove global existence of
(carrying two X chromosomes, XX). solutions as well prove the existence of a finite dimensional
(L2(Ω);L2(Ω)) global attractor to a TYC type model. An
Variations in the sex chromosome number can be immediate mathematical question is: Is it is possible to
produced through genetic manipulation; for example, a improve the regularity of the attractor for such a class of
phenotypically normal and fertile male fish bearing two Y models? Also, from a more practical perspective one might
chromosomes termed supermales (s) [2, 6, 7, 8]. Additional ask, what is the decay rate of the female species?
manipulations through hormone treatments can reverse the
sex, resulting in a feminized YY supermale [36, 33, 22].
The eradication strategy involves the addition of sex-
IJISRT19AP407 www.ijisrt.com 569

ISSN No:-2456-2165
In the current manuscript, function Ω is the rate of introduction of the sex reversed
supermale. These coefficients are all allowed to be
 We consider a major modification to the model in [46] functionally dependent.
by considering the introduction of both supermales and
feminized supermales. Note in [46] we were only able Equation (4) is independent of the three first
to show that the global attractor of the considered equations. It is the heat equation under homogenous
system was an (L2(Ω);L2(Ω)). In this manuscript, we Dirichlet boundary conditions. Under standard conditions
show that the attractor is actually a (L2(Ω);H2(Ω)) on the reaction term r4:
attractor, thus improving the results in [46].  ²[   r   g₁  r  r ] 
dr
 We show that extinction is always possible, under  0 and   ,
certain parameter restrictions, via the proposed strategy,
r ² r0
  r   g₁  r  r (8)

even in a population which is not governed by a logistic where r0  max r0 ( x).
type control term. The attractor is seen to be a one point x
attractor.
 We perform numerical simulations to investigate the see []. The solution of (4) with the given boundary
decay rate of the female species, showing numerical conditions exists globally in time and is bounded on
evidence of exponential attraction. We also explore
optimal control scenarios for extinction of the invasive r (t,.)   r (t ), in R , (9)
species, for different reaction type terms.
II. THE MATHEMATICAL MODEL where r (t ) is a bounded function on bounded subsets of
The control method described above is modelled via R+. The primary difficulty to prove the global existence of
the following system of reaction diffusion equations: a solution to (1)-(4), is that the reaction terms given by (7)
can change sign, and thus the solutions to (1)-(3) are not
t f-d1f=r1 (f,m,s,r), (1) bounded a priori.
t m-d 2 m=r2 (f,m,s,r), (2)
III. NOTATIONS AND PRELIMINARY
t s-d3 s=r3 (f,m,s,r), (3) OBSERVATIONS
t r-d 4 r =r4 (f,m,s,r), (4)
in R+Ω with the boundary conditions For the definition of a strong solution we give the
f=m=s=r=0 on R+Ω (5) following (see for example [29])
Definition. 3.1. We say that

where  is an open bounded domain in Rn, n=1,2,3 with
u (t,. )=: [0,T L2()L2()L2()L2() , is a strong
smooth boundary . The functions f, m, r and s are the
population densities of the normal females, normal males, solution of the system (1)-(4) if:
supermales and sex reversed supermales respectively. The i) u is continuous on [0,T [ and u (0,.)  u0 (.).
constants d , d , d and d are positive, called diffusion ii) u is absolutely continuous on compact subsets of
1 2 3 4
coefficients. The functions gi, i=1,...,10 and  ]0,T [.
iii) u is differentiable on ]0,T [.
are polynomials with positive coefficients. The initial data.
f(0,x)=m(0,x)=s(0,x)=r(0,x)=0, in Ω, (6)
We say u (t,. ) is classical if it satisfies (1)-(4) pointwise, in
are assumed to be nonnegative and uniformly bounded on
. The reaction terms are given by: the usual sense of derivatives. That is we require, .
Our aim is to construct polynomial Lyapunov functionals

 1  (see S. Kouachi and A. Youkana [25] and S. Kouachi [26,
r1 (f,m,s,r)= 2 g1 (f,m,s,r)fm-g 2 (f,m,s,r)f, 
  27]) involving the solutions (f,m,s) of system (1)-(3), so
r (f,m,s,r)= 1 g (f,m,s,r)fm+g (f,m,s,r)fs  that we may estimate their Lpbounds and deduce global
2 2
3 4

  (7) existence.
+ 1 g (f,m,s,r)mr-g (f,m,s,r)r, 
 5 6 
The usual norms in spaces Lp(), L() and
 2 
 1 
C () are
r3 (f,m,s,r)= 2 g 7 (f,m,s,r)mr+g8 (f,m,s,r)rs-g 9 (f,m,s,r)s,  respectively denoted by
 
r4(f,m,s,r)= (r )  g10 (r)r 
  p p
  1

|| 
∥u∥ = | u(x)| dx, (10)
p 
Here g , g , g , g , g ,g are the mating rates, and
1 3 4 5 7 8 and
g2, g6, g9, g10 are the death rates, of the species. The

ISSN No:-2456-2165
u 
 max u ( x) . (11) that the reaction terms are locally Lipschitz (see for
x example [9]).
Since the nonlinear right hand side of (1)-(4) is IV. GLOBAL EXISTENCE
4
continuously differentiable on R+, then for any initial data
For the global existence of the system (1)-(3), we
in C () or Lp(), p 1,+ ( )
, it is easy to check introduce the following functional used in S. Kouachi []
directly its Lipschitz continuity on bounded subsets of the Lp (t )   H p ( f (t , x), m(t , x), s(t , x))dx, (14)
domain of a fractional power of the operator 
where
d1 0
 
0 0 p q q i
i qi pq
H (f,m,s)=   C C   f m s
 0 d  0
2
0
. (12)
p
q=0 i=0
p q i q
. (15)
 0 0 d3 0
 The sequences  and 
{ } { }
are real and
 0 0 0 d4 
i
iN
q
qN
It is well known that to prove global existence of solutions positive satisfying
to (1)-(3) (see, for example [19]), there are several methods
such as the method of comparison with corresponding ii  2 2
ordinary differential equations, method of invariant regions  d3, i  1,..., q, (16)
and functional methods based on a priori estimates. This
i 1
2
last method, implies in several cases the global existence in and
 
time by application (to the reaction terms) of the well   q q  2 2   2 2 2 2
known regularizing effect (see for example [9]) which is  2  d 1   i 2i  2  d 3   d 2  d 1 d 3 ,
also called LpL smoothing effect of the heat operator   q 1   i 1  (17)
(i.e. the diffusion equation has an instantaneous i  1,..., q, q  1,..., p,
regularizing effect in the sense that the above solution u
belongs to L  0,Tmax ,L ()  regardless of the
[ [ where
 
di  d j
regularity of the initial data and that of the reaction to dk  , i  j  k, i, j , k  1, 2,3.
belong to L  0,Tmax ,Lp ()  for some p>N/2). The
[ [ 2 di d j
 
proof is based on the Riesz-Thorin interpolation Theorem (18)
(see e.g. [11]). Rigorously it suffices to derive a uniform Remark 3 Conditions (16)-(17) imply that the sequences
estimate of each ri ( f , m, s, r , , 1i4 on [0,Tmax  i+1  q+1
p [    and    are increasing and the
for some p>N/2 and deduce that the solution to (1)-(3) is in  i iN  q qN
 sequences i
[
L () for all t 0,Tmax , where Tmax denotes the [ { } and {q} and can be chosen as
iN qN
 follows
eventual blow-up time in L (). Under these assumptions,
   
the following local existence result is well known (see i2 q2
[19,12,51,and 58]). i  K d k 3 and  i  K d 1 ,
(19)
Proposition 1 The system (1)-(4) admits a unique, classical i, q  0,1,...,
solution (f,m,s,r) on [0,T [ . Furthemore if where K and K are any positive constants.
max
Tmax< then We suppose that the polynomials g , g , g and g
2 6 9 10
lim  
f (t ,.)  m(t ,.)  s (t ,.)  r (t ,.)   (13) (not all constant) are sufficiently large, that is in term of
   
t Tmax limits

1
g3 fm  g 4 fs  12 g5mr
where T
max
denotes the eventual blow-up time in L (). lim 2
 ,
f msr g2 f
Remark 1 In our setting a classical solution to (1)-(4) can and (20)
be proved to be a strong solution. However, we refrain
2 g6
from this at present time.
lim  ,
f msr g1 f
Remark 2 The uniqueness of the solution which is a fixed and
point of a nonlinear operator, is obtained by using
standard arguments (Fixed Point Theorem) and the fact

ISSN No:-2456-2165
g7 mr  g8rs  g9 s
1 where the three order matrices Biq are given by
lim 2
 , (21)
f msr 2 g6 r  2 g3 fm  g 4 fs  2 g5 mr
1 1 1
 a q  2i  2  a 2b   q  2i 1  a 2c   q 1i 1 
 
Biq =   a 2 b   q  2i 1 b q  2i  b2 c   q 1i 
or
g7 mr  g8rs   ac    b2 c   q 1i c qi 
1
2
 ,  2 q 1 i 1 
lim
f msr g 2 f  g6 m
(22)
(26)
0  i  q, 0  q  p  2,
and
and T denotes the transpose vector
g8 s
 , T   f , m, s  .
t
lim
f msr
1 i1
2 i g1 fm  2 g3 fm  g 4 fs  2 g5 mr
1 1
(23)
From Sylvester’s criterion, each of the quadratic forms
Remark 4 Conditions (20), (21) and (22) imply that the (with respect to f, m and s ) associated with the
 i+1 matrices Biq, 0qp2, 0iq is positive, if we prove the
intervals in which we choose the sequences   
 i iN positivity of its main determinants
 q+1 iqj , j  1, 2,3. For a fixed 0iq and 0qp2, we see
and   become sufficiently large and this gives
 q qN
that
1
us more freedoom to choose the sequences. iq=d1q+2i+2>0,
Remark 5 Also note, the gi(f,m,s,r) , i=2,6,9,10, and condition (16) implies
cannot all be chosen as constant. This will violate (20),
(21) and (22). Note, if the gi(f,m,s,r) are all 2 2 2  ii+2 
iq=d1d2q+2i+1   d3 2 >0.
chosen to be constant, then for certain data   2 
 i+1 
(f0,m0,s0,r0)L() (possibly large) the solutions to We can show by elementary calculation that
problem (1)-(4) can blow-up in finite time. We demonstrate
this via numerical simulation. See [30] for theoretical
results on blow-up for similar systems. Also see [31] for a
 
iq2  d1d 2 d3 q21i21i . i . q  d 2  d 1 d 3 
2 2 2
 
blow-up approach to controlling invasive populations. i  1,..., q, q  1,..., p,
Thus we can state the following result where
Theorem 4.1 Let ( f (t ,.), m(t ,.), s (t ,.), r (t ,.)) be any  ii  2 2   q q  2 2
i    d3 , q   d1 
positive solution of the problem (1)-(4) and suppose that  
   
2 2
the polynomials g2, g6, g9 and g10 are sufficiently large i 1  q 1 
3
(conditions (20)-(23)), then under conditions (16)-(18) the and this gives from (17)  >0. Consequently we have I0.
functional Lp(t) given by (14) is decreasing on the interval iq
For the second integral we have many ways to prove that
[0,Tmax[ . J0, but we choose only two ways:
Proof. The first, since
Following the same reasoning as in S. Kouachi [28], that is p 2 q
by differentiating Lp with respect to t we got I   p( p  1)  Cqi C pq2 iq f i mq i s p 1q dx,
 q 0 i 0
'  f m s
Lp(t)=
 fHp t+mHp t +sHp t dx where
   q 1  i 1  
 iq    r1  r2   r3   qi
=
 (afHpf+bmHpm+csHps)dx (24)   q  i  
 with i  1,..., q, q  1,..., p .
+
 (r1fHp+r2mHp+r3sHp)dx Replacing the reactions r1, r2 and r3 by their respective
 values given by (7), we get
=I+J.  iq  q 1
Using Green’s formula and the boundary conditions via(5),  Gi  12 g 7 mr  g8 rs  g9 s,
we obtain  qi  q
p 2 q
I   p ( p  1)   [C pq 2Cqi  BiqT  .T ] f i m q i s p  2q dx,
where
i 1 1
 q 0 i 0 Gi   g fm  g2 f   12 g3 fm  g 4 fs  12 g5mr  g6m
(25) i 2 1
(27)

ISSN No:-2456-2165
Then J0, if we choose first i  0,1,..., p,
Gi  0, where C1 is a positive constant depending on p.
If we suppose that the reaction terms are of polynomial
which can be satisfied if we choose growth
i+1 i+1
gi ( f , m, s, r)  C2 ( f , m, s, r )[1  f  m  s  r ]l on R3 , (29),
g1fmg6m<0< g f g312fmg4fsg512mr,
2  2
i i i  0,1,..., p,
and also this is satisfied if where C2 positive and bounded function on bounded
4
12g fm+g fs+12g mr  2g m subsets of R+ we have the following
3 4 5 i+1 6
  ,
g f
2  i
g fm
1 Proposition 2 If the reaction terms are of polynomial
growth with g2, g6, g9 and g10 are sufficiently large, then
then under condition (20), we can choose the sequence
 
{ } i
satisfying (16). Secondly by choosing the all positive solutions of (1)-(4) with initial data in L ()
iN are global.
sequence q { } satisfying Proof: From corollary 1, there exists a positive constant C
3
qN such that
 q 1 1
 2 g6m  12 g3 fs  12 g5mr  g4 f   12 g7 mr  g8rs  g9 s  0,  (1  f (t, x)  m(t, x)  s(t, x)) dx  C ,
p
 3 on [0, Tmax [,
q 
(30)
for all p1 and from (26) we have
which can be chosen under condition (21).
ri  f , m, s, r  l2  C2  f , m, s, r  (1  f  m  s  r ) p ,
p
The second way is that we choose on [0,Tmax [,
(31)
q+1  i+1  * p [ [
 Since f,m,s and r are in L ( 0,T ;L ()), for all
q  i
g112fm+g312fm+g4fs+g512mrg9s p N
  p1, then we can choose p1 such > and from the
l+2 2
< preliminary observations the solution is global.
q+1  i+1  Remark 6 The global existence can be proved under more
q  i 2
0< g f +g6mg712mrg8rs, general boundary conditions including homogeneous and
  nonhomogeneous Dirichlet, Neumann and mixed boundary
conditions (see [28]). Also note, because the non linear
that is semi-group S(t) in this case is regularizing [59], for initial
data say u L2(), for some r>0, S(r)u Lp(). We can
0 0
1
g7 mr  g8rs q 1 g9 s
2
  . now use the constructed functional (14) with initial data
i 1   S(r)u0, which is in Lp(), so the local solution is in Lp(),
g f  g6 m q 1 i 1
g fm  2 g3 fm  g 4 fs  2 g5mr
1 1
i 2 2
i 1 thus can’t blow up and becomes global. Thus we have a
 p
priori L (0,;L ()) bounds for data in L ().
2
As the gi 's are polynomials with positive coefficients, then
condition (22) together with (23) permit us to choose the V. BOUNDED ABSORBING SETS AND FURTHER A
sequence q PRIORI ESTIMATES
{ } satisfying (17). This ends the proof of
qN
5.1 Bounded absorbing sets
the Theorem. By application of the preliminary
In this section we aim to investigate the asymptotic
observations, we have the following,
behavior of (1)-(4). We use the functional Lp to show the
Corollary 1 Suppose that the reaction terms are
4 existence of bounded absorbing sets. Using the fact that the
continuously differentiable on R+, then all positive
matrices Biq, i= 0,q ,q= 0,p are positive definite,
solutions of (1)-(4) with initial data in Lp() are in we can find a constant C4 such that
L(0,T ;Lp()) for all p1.
max  p2 2
Proof: If p is an integer, the proof is an immediate Ip(p1)C 
4 (
df+m) | (f+m+s)| dx, (32)
consequence of Theorem 3.1 and the trivial inequality

 ( f (t, x)  m(t, x)  s(t, x)) dx  C L(t),
p
1 on [0, Tmax [, (28) and this gives


ISSN No:-2456-2165
L'p (t ) Remark 8 Typically, in order to make dissipative

estimates, we require an inequality of the form
2
 p( p  1)C4  ( f  m  s) p 2 ( f  m  s) dx
d
(33) ||u||V+C2||u||VC1, for a state variable u in some
dt

function space V. The C ,C are pure constants, that could
 0, 1 2
depend on the parameters in the problem, but not on the
on [0, Tmax [, initial condition [50]. Methods in [50] show that typically
by integrating (33) with respect to t, we deduce 1
if we choose t = ln(||u || ), then for times t>t , we have
1 C 0 V 1
2
 ( f  m  s) dx
p
C
1
 that ||u|| 1+ . In our setting the R.H.S does depend on
V C2
t (34)
C5  ( f  m  s) p 2
2
( f  m  s) dxds  Lp (0), the initial condition, however we can give an (,)
2
0 argument to show that the L () norm of the solutions is
on [0, Tmax [, still absorbed by a finite time t1. Note in the estimates it is
then this inequality gives assumed conditions (20)-(23)), and conditions (16)-(18)
hold.
Via the use of Gronwall’s lemma [50] in (38) we obtain
f,m,sL [0, ;L
( ) (
[ p () L2 [0, ;H [ 1 () . ) C
||f|| e2t||f || + (||f || +||m || +||s || )(1e2t). (39)
(35) 2 2 1 2 2 2
The above method shows the existence of bounded 2 02 2 0 2 02 0 2
absorbing set in Lp(), for all p1, and so in particular for Note for any 2>>0, there exists a t=T*(), s.t
L2(). Similar estimates are made in [48]. e2t1 t

=e . Thus for t[0,T*()] , we have that
' 2t
e
Remark 7 Note, from (33) it is immediate that L (t)<0,
2
e2t1 t
hence L2(t) is decreasing in time. From the form of the e .
e2t
2
functional Lp(t) in (14)-(15), it is clear that ||f||2 is also 1 2 2 2
Case 1: ln(||f0||2+||m0||2+||s0||2))T*().
decreasing in time, and must enter some compact ball, by a 
2 2 2
finite time t1, where t1 will depend on the L2() norm of We assume ||f0|| 2+||m0||2+||s0||2>1, else the absorbing set
the initial conditions, and the parameters in the system. is trivial from (34).
For completeness we show certain details pertaining to the e2t1 t
Using the fact that e for t[0,T*()]
uniform L2() estimates. e2t
Let us begin by multiplying (1) by f, and integrating by
parts over , to obtain
on [0, Tmax [, , in (39) we obtain,
1d 2 2  +2  +1  +2  2 C1
||f|| +||f||2=a1  f 1 m 1 dxa2  ||f||2e2t||f 0||2+ (||f0||2+||m0||2+||s0|| 2)et.
2 2 2 2
2 dt 2  f 2 m 2dx. 2
(40)
  1 2
(36) Let’s choose t0= ln(||f0||2). Also given (f0,m0,s0) an >0,
2
We now use the positivity of f and m along with Holder’s
we can find a >0 s.t
inequality to obtain
1+2 1+1 1 1 T*()
1d 2 2 < < .
||f|| 2+||f||2a1||f|| ||m|| C. (37)   2 2 2
2 dt 2(1+2) 2(1+1) ln(||f || +||m || +||s || )
02 0 2 02
This follows via the a priori Lp bound on the solutions, (41)
and hence in particular for p=max(2(1+2),2(1+1)) . *
Now we choose t1 such that
2 * 1 2 2 2
Note here C only depends on the L () norm of the initial t = ln(||f || +||m || +||s || )),
data, and is independent of time. The C here comes from 1  02 02 0 2
2 2 2 (42)
L2(0), which can be bounded by C1(||f0||2+||m0|| 2+||s0||2),
Note, given 2>>0, we can always find >0 via (41), so
where C1 is a pure constant. Thus we use Poincare’s *
that t <T*().
inequality to obtain, 1
1d 2 2 2 2 2 *
||f|| +||f|| C (||f || +||m || +||s || ). (38) Finally we choose t =max(t ,t ), then we have
2 dt 2 2 1 02 02 02 1 0 1

ISSN No:-2456-2165
C1
f 2  f
1 d 2 2
2
||f||21+ ,  t>t1. 2 dt 2
2
(43)  a1  f 1 1
m 1 1
(f )dx  a2  f 2 1m 2 (f )dx.
1 2 2 2  
Case 2: ln(||f || +||m || +||s || ))>T*(),
 02 0 2 02 (52)
This might be the situation if for example the given initial
data (f ,m ,s ) was very large. Then similarly as earlier we
0 0 0 Then employing Young’s inequality yields,
1d 2 2
2 C1 ||f|| +||f||
have ||f||2e2t||f 0||2+ (||f0||2+||m0||2+||s0|| 2)et. (44)
2 2 2 2 2 dt 2 2
2 4(1+1) 4(1+1)
1 2
Also given (f ,m ,s ) an >0, we can find a >0 s.t  ||f|| +2||f|| +2||m||
0 0 0 4 2 4(1+1) 4(1+1)
*
1 T () 4(2+1) 4(2)
< , 0<<. (45) 1 2
 2 2 2 + ||f||2+2||f|| +2||m|| ,
ln(||f0||2+||m0||2+||s0||2) 4 4(2+1) 4(2)
Thus which via the a priori Lp bounds on the solutions, hence in
C particular for
||f|| e2t||f || + (||f || +||m || +||s || )et.
2 2 1 2 2 2
(46) p=max(4( +1),4( +1),4( +1),4( )) , leads
2 02 2 0 2 02 0 2 1 1 2 2
Now we choose to
* 1 2 2 2 4(1 1) 4(  1)
f 2  f 2  4 f  4 m 4( 1 1)
d 2 2
t1= ln(||f0||2+||m0||2+||s0|| 2)), (47)
 dt 4(1 1) 1
Thus given 2>>0, we can always find >0 via (45), so 4( 2 1) 4 2
* 4 f 4( 2 1)
 4 m 4   C.
that t1<T*(). 2
(53)
* 2 1
Finally we choose t1=max(t0,t1), and we have Now using the Sobolev embedding of H ()↪H0(), we
C obtain
2 1
||f||21+ ,  t>t1. (48) d 2 2
2 ||f||2+C1||f||2C.
1 dt
We next demonstrate next the H0() estimates with f. (54)
We integrate (37) in the time interval from [t,t+1] for any Grönwall Lemma via integration in the time interval [t,t]
tt1, to obtain yields the following uniform bound
 CC1 1  e (t t *)   e (t t *) f  t * 2
2
f
t+1 2
2 2 2
||f(t+1)||2+ 
2
 ||f|| 2dsC5+||f(t)||2C6, tt1. (49)
t  CC1  C6 , t  t*  t1.
C1 (55)
Remark 9 Note the C6 absorbs C5 and 1+ from (48),
2
This follows via (51).
C1
so 1+ +C5<C6. .
2
Remark 10 Note, (51) holds for any t>t*. However, the
Thus we have the following uniform integral in time bound *
t+1 reason we use t , is to first derive a uniform in time bound

2 
 ||f||2dsC6, tt1, on the ||f(t)|| 2 via (51), so that the e(tt ))||f(t)|| 2
(50) 2 2
t
using the Mean Value Theorem for integrals, there term can be absorbed, uniformly in time for times tt t1,

*
exists t [t,t+1] such that for all t>t1, we obtain using (51).
* 2
||f(t )||2dsC6, (51) 1
Remark 11 The strategy for the uniform H0() estimates
We next multiply (1) by f and integrate by parts over .
For such higher order Sobolev estimates, we will assume f for the m,s,r components is similar. That is we can derive a
m 2 m
and f satisfy the same boundary conditions, and similarly finite time t3 s.t. ||m||2C, for t>t3 . Here the finiteness
the same is true for the other components. Thus we obtain
m
of the time t , comes via the methods similar to (36)-(55),
3
where we use the equation for m via (2). Similarly we can

ISSN No:-2456-2165
s 2 s
derive a finite time t s.t. ||s|| C, for t>t , and we can d1
3 2 3 2 4(2+1) 4(2)
r 2 r + ||f||2+2||f|| +2||m|| ,
derive a finite time t s.t. ||r|| C, for t>t . 4 4(2+1) 4(2)
3 2 3
This leads us to state the following Lemma. p

which via the a priori L bounds on the solutions, hence in
particular for
Lemma 5.1. Let f,m,s be solutions to (1)-(4) with p=max(4(1+1),4(1+1),4(2+1),4(2)) leads to
2
(f ,m ,s ,r )L (). Assume conditions (20)-(23) and
0 0 0 0 d
f 2  d1 f 2
2 2
1
conditions (16)-(18) hold, and the finite H () absorption dt (65)
0
4( 1) 4(  1) 4( 2 1) 4 2
m s r  4  f 4(1 1)  m 4( 1 1)  f  m 4    C.
times for the components f, m, s, r are t1,t3 , t3 and t3  1 1 4( 2 1) 2 

respectively. We denote t by
3 Note the regularizing properties of the semigroup yield Lp
 m s r 2
t =max(t1,t3 ,t3,t3). (p>2) bounds on the solution, for initial data in L . We now
3

integrate (65) above from t to t to obtain
There exists a constant C independent of time and initial 3
data, and depending only on the parameters in (1)-(4), such t t
2
  ||f||2ds 
  Cds.
that for any t>t the following uniform a priori estimates
3  
hold: t t
3 3
In particular choosing
2  (56)
||f||2 C, (56) t=t +1,
3
2
||f|| C, (57) yields (57)
2 
2 t +1
||m||2 C, (58)
1
3
2
(58)
2  

 |f|2dsC.
||m||2 C, (59) (t +1)t (59)
3 3 t
2 3
||s||2 C, (60) (60)
2 Hence using a mean value Theorem for integrals we obtain
||s||2 C, (61) (61)
  
2 that there exists a time t t ,t +1] such that the following
3 3 3
||r||
2
C, (62) (62)
estimate holds uniformly
2  2
||r|| C. (63) ||f(t )|| C. (63)
2 3 2
5.2 Local In Time a Priori Estimate for f. Remark 12 The strategy for the uniform integral in time
Our goal now is to show that we can derive a priori
H2() estimates for the m,s,r components is similar. That
2
H () bounds on the solutions to (1)-(4). To this end we m
next multiply (1) by f and integrate by parts over  to is we can derive a finite time t3* s.t.
obtain m
t3*+1
f 2  d1 f  a1  f 1 1m 1 1 (f )dx
1 d 2 2
2 m
2 dt 2


 ||m||2dtC, for t>t3*. Similarly we can derive a
(64) m
a2  f  2 1 2
m (f )dx, t
3*
 s
finite time t s.t.
3*
Then employing Young’s inequality yields, s
1d 2 2 t3*+1
||f|| +d ||f|| 2 s
2 dt 2 1 2 
 ||s||2dtC, for t>t3*,
d1 2 4( 1+1) 4(1+1)
 ||f|| +2||f|| +2||m|| s
4 2 4(1+1) 4(1+1) t
3*

ISSN No:-2456-2165
r 2 d1 2
and we can derive a finite time t s.t.
3* ()=|f|2, ()=C3 , h()=C2+C4|m|2, (71)
r 2
t3*+1 and application of the above lemma yields
2

 ||r||2dtC. Lemma 5.3. Let f,m,s be solutions to (1)-(4) with
r 2
t (f0,m0,s0,r0)L (). Assume conditions (20)-(23) and
3*
conditions (16)-(18) hold, and we have the finite integral in
2 2
5.3 Uniform A Priori H Estimate For f. time H () estimates for the components f, m, s, that is,
We multiply Equation (1) by 2f and integrate by
*
parts over  to obtain t3+1
2 2 *
d||f||
2 2 
 ||f||2dtC,  t>t3
+d ||(f)||
dt 1 2 *
  +1  +1  t3
  (a f 1 m 1 a2f 2 m 2)(f)dx=
+1
 1  m
  t +1
3*
2 m
 ||m||2dtC,  t>t3*

  +1   +1 
 (a1f 1 (1+1)m 1m+a1m 1 (1+1)f 1f)(f)dx t
m
 3*
s
t3*+1
 +1  1  

 (a2f 2 (2)m 2 m+a2m 2(2+1)f 2f)(f)dx 
2 s
 ||s||2dtC  t>t3*
 s
d1 t3*
2 2 2
C2+ ||(f)||2+C3||f||2+C4||m||2. (66) (66)
2 
We denote t by
4
This follows via the a priori Lp() bounds on the
 * m s
t =max(t3,t3*,t3*).
4
solution, the embedding of H2()↪W1,4() , Cauchy-
Schwartz and Young’s inequalities. Now using the Then there exists a constant C independent of time and
embedding of H3()↪H2() we obtain, initial data, and depending only on the parameters in (1)-

(4), such that for any t>t the following uniform a priori
2 4
d||f||2 d1 estimates hold:
2 2 2
+ ||f||2C2+C3||f||2+C4||m||2. (67)
dt 2

||f(t)|| 2
H ()
C, tt4+1,
Now we recall the Uniform Grönwall Lemma

||m(t)||H2() C, tt4+1,
Lemma 5.2. (Uniform Gronwall Lemma) Let , and h
1 
be nonnegative functions in L [0, [. Assume that  is ||s(t)|| 2
H ()
C, tt4+1.
loc
absolutely continuous on ]0, [ and the following Thus the existence of a bounded absorbing set in H2() has
differential inequality is satisfied. also been established.
d
+h, for t>0. (68) f
dt 5.4. Uniform a Priori Estimates for
If there exists a finite time t1>0 and some r>0 such that From (1) via brute force we obtain t
2
t r t r t r
  f
    d  A,
t
    d  B
t
and  h   d  C ,
t
(69)   t2
2
d1f+a1f1+1m1+1a2f2+1m2
for any t>t , where A, B and C are some positive constants,
1 =
 ( ) dx
then

(t)  r +CeA, for any t>t1+r.
B
(70) 2 4(1+1) 4(1+1)
C||f||2+C1||f|| +C2||m||
Thus using 4(1+1) 4(1+1)

ISSN No:-2456-2165
2 2 2  
C||f|| +C ||f|| +C ||m|| . (72) (72) interval [t ,t +1]
Thus integrating the above in the time
2 3 2 4 2 6 6
yields
This follows via the priori Lp() bounds on the t6* 1
 f 2

solution, as well as the compact embedding of   ds 
H2()↪Lp(), p (since the spatial dimension n3). t6*  s 2
Similar estimates can be derived for the m,s components.

1 ( 1 1)
t6* 1
 m 2 f 2
 1 ( 1 1)  f 2 
t6* 1
We can now state the following Lemma, C f m    ds  C f m   s 2 ds
 s 2 s
 
t6* 2 t6*  
Lemma 5.4. Consider (1)-(4), for any solutions f,m,s and r
of the system with (f ,m ,s ,r )L2(). Assume conditions ( 2 1) (  2 1)
t6* 1
 m 2 f 2
 2 2
t6* 1
 f 2 
0 0 0 0

C f 
m  
s

s
ds  C f 
m *  s 2 ds.
(20)-(23) and conditions (16)-(18) hold. We denote t by t6*  2 2 t6  
6
  1 ( 1 1)
t6* 1
 m 2 f 2

   ds
* *
t =t +1. C f (t ) m(t )
 s 2 s
6 4 6 H 2 () 6 H 2 ()
t6* 2
Then there exists a constant C, independent of time and

initial data such that the following estimates hold uniformly
1  f 2  ( 1 1)
t6* 1
H (  )   s
f C f (t ) 2  ds
2 * *
m(t ) 2
 C,   t6* , H () 6 6
t 2
t6*  2
m t6* 1
 m 2 f 2 
2
 C,   t6* , * ( 2 1) * (  2 1)

C f (t6 ) 2 m(t6 ) 2   
 s 2 s 2 
t ds
H () H ()
t6  
2
*
s
2
 C,   t6* , t6* 1
 f 2 
t * 2 * 2
2
C f (t ) m(t )   ds. (73)
 s 2 
6 H 2 () 6 H 2 ()
r
2
t6*
 C, t *
t
6
2  C.
This easily follows via the estimates in Lemma 5.5. We
f This follows via the regularizing properties of the
next make a local in time estimate on We take the
t semigroup, Lemma 5.5, Lemma 5.4 and the embedding of
H2()↪L()and hence using a mean value Theorem for
partial derivative w.r.t t of (1) and multiply the resulting
f
equation by and integrate by parts over  to obtain integrals we obtain that there exists a time
t
1 d f
2
f
2 t6
**
 t6 , t6  1
* *
such that the following estimate holds
 d1  uniformly
2 dt t 2 t 2 2
f (t6** )
 C.
   +1  m   +1 f  f t
 a1(1+1)f 1 m 1 t +a1(1+1)f 1m 1 t  tdx 2

f
2
We will next make a uniform in time estimate for ,

t
  +1  1 m   f  f 2
+ a2(2)f 2 m 2 t a2(2+1)f 2m 2 t  tdx where the previous estimate will be used. We take the time
 f
derivative of (1), then multiply through by  and
 t
 m 2 f 2  integrate by parts over  to obtain
( 1 1)  f 
2
1 ( 1 1) 1
C f  m    C f  m  
 t 2 t 2   t 2 
   
(  2 1)  m f  (  2  f 
2 2 2
2 2
( 2 1) 2 1 d  f f
C f  m  
 t 2 t 2   C f  m  
 t 2 
. +d   
2 dt  t  1   t
    2 2

ISSN No:-2456-2165

 m f   f  f
2

a1   ( 1  1) f 1 1m 1  (1  1) f 1 m 1 1     dx C, tt6 ,
 t t   t  t L2 (  )
 f   f  m
2
m 
a2    2 f 2 1m 2 1  ( 2  1) f 2 m 2     dx C, tt6 ,
 t t   t  t L2 (  )
r
2


C, tt6 ,
t

 m 2 f 

2 L2 (  )
2(1 1) 2( 1 1) 2( 2 1) 2 2
C max f m , f m    s
2

 t 2 t
 
2 C, tt6 .
t L2 (  )
d1 f
2
 .
2 t 2 VI. EXISTENCE OF GLOBAL ATTRACTOR
In this section we prove the existence of a compact

This follows from the product rule for differentiation, global attractor for system (1)-(4).
Cauchy-Schwartz inequality and the Sobolev embedding of
1 6.1 Preliminaries
H0()↪L4(). Now using the embedding of Recall the phase space H introduced earlier
1
H2()↪H0() we obtain
H=L2()L2()L2()L2().
Also recall
2 2
f f
d
dt ∥ ∥ ∥ ∥
t
2
+d1
t
2
1 1 1 1
Y=H ()H ()H ()H (),
0 0 0 0
K K  f
2
m
2 and
C||f|| 2||m|| 2 ∥ ∥ ∥ ∥ 
X   H 2 () H 01 ()    H 2 () H 01 ()  
+
 t t
H H 2 2
C.
H 2
() H 01 ()    H 2 () H 01 () 
Here K=max(2(1+1),2(1+1),2(2+1),2(2)) .
 Recall the following definitions
Thus via time integration in the interval [t ,t] in the
6
Grönwall Lemma we obtain Definition 6.1. Let AH2(), then A is said to be a (H,X)
f  t6**  1  e d1 (t t6 )
2 global attractor if the following conditions are satisfied
f
2 **
 d1 ( t t6** ) i) A is compact in X.
e  C  C. (74)
t 2 t d1 ii) A is invariant, i.e., S(t)A=A, t0.
2 iii) If B is bounded in H, then
t  t ** dist (S(t)B,A)0, t.
6 . X
We can make similar estimates for the other components,
m s r Definition 6.2. (Asymptotic compactness) The semi-
and derive similarly absorbing times t ,t ,t , where group {S(t)} associated with a dynamical system is said
6 6 6
m s r t0
(t ,t ,t ) are the absorption times for
6 6 6 2
to be asymptotically compact in H () if for any
m(t ) r (t ) s(t ) 
, , . 2
t L2 (  ) t L2 (  ) t L2 (  ) { }n=1 bounded in L () and a sequence of times
f0,n
{tn}, S(tn)f0,n possesses a convergent subsequence

We thus state the following result,
in H2().
Lemma 5.5. Consider (1)-(5). For any solutions u,v,w,z to
Definition 6.3. (Bounded absorbing set) A bounded set B
the system, there exists a constant C independent of time
in a reflexive Banach space H is called a bounded
  m s r
and initial data, and a time t =max(t ,t ,t ,t ), absorbing set if for each bounded subset U of H, there is a
6 6 6 6 6
time T=T(U) depending on U, such that S(t)UB for all
such that the following estimates hold uniformly,

ISSN No:-2456-2165
t>T. The number T=T(U) is referred to as the which means that ( f , m , s ) is a constant vector.
compactification time for S(t)U. This is the time after
which the semigroup compactifies. Using the homogenous Dirichlet boundary conditions, we
deduce f = m = s =0, on the interval (0, ). This
Also recall that if A is an (H,H) attractor, then in order to end the proof of the proposition.
prove that it is a an (H,X) attractor it suffices to show the Now we can state the following
existence of a bounded absorbing set in X as well as
demonstrate the asymptotic compactness of the semi-group Theorem 6.5 The attractor of the semi group {S(t)} is a
in X, see [60]. We first state the following Lemma.
t0
Lemma 6.4. Consider the system described via, (20)-(23). one point attractor.
Under conditions (1)-(4), there exists a (H,H) global Proof: By direct application of A.V. Babin and M.I.
attractor A for this system which is compact and invariant Vishik [3] [Theorem 10.2, page 2.4], we can deduce the
statement of the Theorem.
in H, and attracts all bounded subsets of H in the H metric.
We next place sufficient conditions on the gi and show
Proof: The existence of bounded absorbing set in H follow
via the estimates derived in Lemma5.1. Furthermore the that in certain special cases, f will decay exponentially to 0,
compact Sobolev embedding of via the following Lemma.
Y↪H
yields the asymptotic compactness of the semi-group Lemma 6.6. Consider the model system (1)- (4). If the
{S(t)} in H. The existence of an (H,H) global attractor reaction term in the equation for r is not positive, then for
t0 2
any constant gi, i=1,3,4,5,7,8,9, and f0,m0L (), we can
now follows.
choose g2,g6, s.t f,m0 , exponentially in the L2() norm.
6.2. One Point Attractor
In this subsection, we shall prove that the attractor is a Proof: If the reaction term on r is not positive, r
one point attractor. We begin by the following trivially goes extinct leading to the extinction of s. This
reduces (1)- (4) to
Proposition 3 The unique fixed point of the semi group
{S(t)} associated with the dynamical system (1)- (4) is tf=d1f+g1fmg2(f,m)f, (75)
t0
the null solution, if the reaction term in the equation for r is
not positive and (20)-(23) hold. tm=d2m+g3fmg6(f,m)m, (76)
with Dirichlet boundary conditions. For the sake of
Proof: Suppose that u =( v , r ) is a fixed point of simplicity let us assume g (f,m)=C f2, g (f,m)=C m2 ,
2 1 6 2
the semi group {S(t)} , where v =( f , m , s ).
t0 where C1,C2, will be chosen later. Then multiplying (75)
by f and (76) by m and integrating by parts over  yields
First by supposing the reaction term in the equation for r is 1d 2 2 4
||f|| +d ||f|| +C ||f|| =g  f2mdx,
2 1 4 1
(77)
not positive, we can deduce easily that r 0. for the other 2 dt 2 1

components we use the Lyapunov functional Lp: from
equation (14)-(15). 1d 2 2 4
2 dt 2 2 2 2 4 3
 m2fdx.
||m|| +d ||m|| +C ||m|| =g  (78)
Since for >0, we have S(t) v =0 on the interval (0, .) 
This gives
Using Poincare’s and Young’s inequality with  we obtain
'
Lp(t)=:I+J=0, forallt (0, .) 2
1d 2 2 4 g1 4 g1 2
Since I0 and J0 on the interval (0, ), then ||f|| +C ||f|| +C ||f||  ||f|| + ||m|| ,
2 dt 2 3 2 1 4 2 4 2 2
2
I=J0,t (0, .) Since each of the quadratic forms
(79)
(with respect to f, m and s ) associated with the 2
matrices B 0qp2, 0iq, given by (26) and appearing
iq 1d 2 2 4 g3 4 g3 2
||m|| 2+C4||m|| 2+C2||m||4 ||m|| + ||f||2. (80)
in the expression of the integral I given by (25) are positive. 2 dt 2 4 2
2
This gives
f=m=s=0,t (0, ,)

ISSN No:-2456-2165
C3 C4 2
compactness in H (), would involve making uniform
Choosing  small enough that is <min( , ),
g1 g3 H3() estimates and then using the Sobolev embedding of
 g1 H3()↪H2(). This will be quite cumbersome, and so is
and C ,C large enough that is C >    ,
1
1 2 1  2 C C  circumvented altogether via the following strategy. We
3 4
 min( g , g ) rewrite (1) as
 1 3 
f
we add up the above to obtain d1f= +a1f1+1m1+1a2f2+1m2. (86)
1d 2 1d 2 2 2 t
||f|| + ||m|| +C ||f|| +C ||m|| 0. (81)
2 dt 2 2 dt 2 5 2 6 2
Defining V=m+f, and C =min(C ,C ), we obtain We will demonstrate that every term on the right hand side
7 5 6 2
1d 2 2 of (86) is uniformly bounded in L (). Thus we obtain that
||V|| +C ||V|| 0. (82) 2
2 dt 2 7 2 f is uniformly bounded in L (), which will imply via
This yields
elliptic regularity the uniform boundedness of f in H2().
||V|| e2C7t||V || ,
2 2
(83) Since this can be done for the other variables, the
2 02
thus we have asymptotic compactness in X follows. To demonstrate this
lim 2 we state the following Lemma
||V|| 0, (84)
t 2
which implies Lemma 6.7. The semi-group {S(t)} associated with the
t0
lim 2 lim 2 dynamical system (1)- (4) is asymptotically compact in X.
||f||20, ||m|| 20 (85)
t t
exponentially. This proves the Lemma. fn
Proof: Let us denote fn(t)=S(t)f0,n and u(tn)= |t=t .
t n
Remark 13 Note we must choose g2,g6 super linear at the
We have that
very least, because choosing them as constant or sub linear
can lead to blow-up in finite time for sufficiently large Error!
initial data, and then no convergence to equilibrium (or 
extinction state) is guaranteed [30]. Via Lemma 5.5 we have for tt
6
Remark 14. In the special case that r goes extinct, which
 f C.
happens if the reaction term in (4) is non positive, s follows  t
suit trivially. Then we are essentially left with a 2 species 2
system for f,m. Special cases of this are tackled in [49] See 
pg.133, example 1.10 and references therein. Although Hence for n large enough tnt and we obtain
6
under certain restrictions on the reaction terms existence of
a global attractor can be proved (via the Simon-
 fn
Lojasiewicz type techniques), convergence to equilibrium is   | C.
another matter. For example, let us (assuming r,s0 )  t  t=tn
2
choose g =2C ,g =C fC ,g =2,g =m . Then upon Also via Lemma 5.1 we have the estimate
1 1 2 2 3 3 6
analyzing the Jacobian we see that if we choose
5 1 1 ||f||2C.
C = ,C = then C = (where C =C C ), what we
1 8 2 8 4 2 4 1 2 
3C C3 Hence for n large enough tnt and we obtain
3 6
obtain is J | * * = , while J | * * = . Since
11 (f ,m ) 2 22 (f ,m ) C
4
||fn||2C.
all constants are positive this says J11J22<0, and standard
These uniform bounds allow us to extract weakly
pattern formation results [64] tell us that there exist convergent subsequences. Thus we obtain
diffusion coefficients d1,d2 for which Turing instability will
occur. Thus the base equilibrium state is driven unstable 1
un(tn)u weakly in H0().
because of diffusion, and one will not have convergence to
the spatially homogenous equilibrium solution. 1
f (t )f weakly in H ().
n n 0
6.3. Asymptotic Compactness of the Semi-ggroup In X Now it trivially follows from the form of the reaction terms,
In this section we demonstrate the asymptotic and the simple algebraic inequality
compactness property. We show calculations for f, the other
variables follow similarly. Showing asymptotic

ISSN No:-2456-2165
||F (f ,m )F(f,m)||C ||f f|| +||m m|| .
( ) Numerical simulations show that this attractor is a one
n n n n 2 n 2
point attractor, see section 7.
Here
 +1  +1  +1 
F (f ,m )=a f 1 (t )m 1 (t )a f 2 (t )m 2(t ) VII. NUMERICAL SIMULATIONS
n n n 1n n n n 2n n n n
Thus from the classical functional analysis theory, see [], 7.1 The basic model
and the compact embedding of We now provide the results of numerical simulations
on (88) -(91). In order to demonstrate the proposed strategy
1
H ()↪L2(), we simulate (88) -(91), under a varied choice of parameters,
0 and function g. When g=1, we have the TYC model,
we obtain without a logistic control term. When g=(1(f+m+r+s)/K))
the classical TYC model [43] is recovered.
2
un(tn)u strongly in L (),
 f=d f+a fmgb f, (88)
t 1 1 1
2
fn(tn)f strongly in L (), tm=d2m+a2fmg+b2fsg+c2mrge2m, (89)
 s=d s+a mrg+b rsge s, (90)
t 3 3 3 3
Fn(f n)F(f) strongly in L2().  r=d r+b r.
t 4 4
(91)
Using these convergent subsequences we obtain In the simulations =[0,], so we are in a 1d spatial
domain. We prescribe Dirichlet boundary conditions. The
fnf strongly in L2(). system is simulated in MATLAB R2014, using the PDE
solver PDEPE. We experiment with various parameters, to
However this implies via elliptic regularity that obtain the spatio-temporal profiles of the solutions.
fnf strongly in H2().
This proves the Lemma.

We can now state the following result
Theorem 6.8. Consider the reaction diffusion system

described via (1)-(4). Under the conditions (20)-(23), there
exists a (H,X) global attractor A for this system which is
compact and invariant in X and attracts all bounded
subsets of H in the X metric.
Proof: The system is well posed via proposition 2,
hence there exists a well defined semi-group {S(t)} for
t0
2
initial data in L (). We already have the existence of an
(H,H) global attractor via lemma 6.4. The estimates derived
via Lemma 6.5 give us the existence of bounded absorbing
sets in X. Lemma 6.7 proves the asymptotic compactness of
the semi-group {S(t)} for the dynamical system
t0
associated with (1)-(4), in X. These results in conjunction
prove the Theorem. Figure 1: We fix x=/2 and follow a trajectory in time for
(88) -(91). The blue is the true trajectory, compared to
Remark 15 Via standard methods [60] we can provide e(0.012)t in green. The clear exponential attraction of
upper bounds on the Hausdorff and fractal dimensions of normal females to the extinction state is observed. The
the global attractor in terms of parameters in the model. To parameters are d1 = d2 = d3 =d4 = 0:001; a1 = a2 = 0:002; a3
derive these estimates we consider a volume element in the = b1 = b2 = b3 = e2 = 0:001; b4=0:05; e3 = 0:03;µ = 0:5. The
phase space, and try and derive conditions that will cause it 2
to decay, as time goes forward. This enables an explicit initial data is taken to be e(x) .
upper bound for the Hausdorff dimension of the attractor
3
 C(ai,bi,i,i) 2
dH(A)   ||+1, (87)
 K1 

ISSN No:-2456-2165
T
1 2
J()=max   (f+m) 2 dt.
0
(93)
We search for the optimal controls in the set U where
U={| measurable, 0<, t[0,T], T}. (94)

*
The goal is to seek an optimal  s.t.,
T
1 2
J( )=max 
*
 (f+m) 2 dt (95)
0
We use the Pontryagin’s maximum princinple to derive
the necessary conditions on the optimal control. The
Hamiltonian for J is given by
1
H=(f+m) 2+ f'+ m'+ s'+ r'. (96)
2 1 2 3 4
Figure 2: Here we consider a superlinear source term
2
=(r)=r . In this case r, will blow up in finite time. We We use the Hamiltonian to find a differential equation of
investigate if this source term can cause a faster decay in the adjoint  ,i=1,2,3,4.
i
the female species, in comparison to a constant  perse.
Surprisingly this is not so. We fix a spatial location and
look at the decay of the trajectory of the female f. Here we 7.3 g=1,(r)=
are comparing (88) -(91) with g=1 to (88) -(91) with g=1
and =r2, and same parameter set as in Fig.1 We  '1  t  =1-1  ma1  b1  -2  ma2 +sb 2  ,
observe that there is a sharper decay in f with a constant ,
 '2  t  =1-1a1 f -2  fa2 +rc 2  e2  -3 ra 3 ,
than with =r2.
 '3  t  =-2 b 2 f -3  rb3 +e3  , (97)
7.2 Optimal Control  '4  t  =-2 c2 m-3  ma3 +sb3  +4 b4 ,

Motivated by the results in Fig. 1- 2 we consider the with the transversality condition gives as
following ODE version of (88) -(91),
1(T)=2(T)=3(T)=4(T)=0 (98)
f Now considering the optimality conditions, the

=a1fmg-b1f,
t Hamiltonian function is differentiated with respect to
m control variable  resulting in
=a 2 fmg+b 2 fsg+c 2 mrg  e2 m,
t H
s =  (99)
=a 3 mrg+b3 rsg  e3 s,  4
t
r Then a compact way of writing the optimal control  is
=  r  -b 4 r
t *
(92)  (t)=max(0,4) (100)
(92) *
We want to compare the following 3 cases: Theorem 7.1 An optimal control  U for the system (92)
with g=1,(r)= that maximizes the objective functional J
1. Case 1. g=1,(r)=;
is characterized by (100).
f+m+s+r
2. Case 2. g=1 ,(r)=; f+m+s+r
K 7.4. g=1 ,(r)=;
K
3. Case 3. g=1,(r)=r2.
We will use optimal control theory to illustrate which

strategy is better for the eradication of wild females and
wilde males. Here, consider the control problem

ISSN No:-2456-2165
 fma1   fma2 mrc2  *(t)=max(0,r2 )
 '1  t  =1-1  gma1  -b1  -2  gma2  +gsb 2 
K 
4
 K   K
(108)
 mra3 rsb3 
+3  gma2 
K 
+ , (101.a)
 K Theorem 7.3 An optimal control *U for the system (92)
2

 '2  t  =1-1  fga1 
fma1   fma2
-2 
fsb mrc2
 fga2  2 

+grc 2  e2 
with g=1,(r)=r that maximizes the objective
 K   K K K  functional J is charcterized by (108).
 mra rsb 
-3   3
 gra3  3  ,
 K K  7.6 Numerical Results for Optimal Control
In this subsection, we numerically simulate the model
(92) with the 3 different cases. The following parameters
fma1  fma2 fsb2 mrc2  will be used in the simulation,
 '3  t  =1 +2    fgb2 
K  K K K  a1=a2=a3=b2=c2=0.0045;b2=b3=0.009;b4=e2=e3=0.12.
 mra3 rsb3  Firstly, we set the initial population as follows
 3    grb3  e3  , (101.b)
 K K  f =20,m =20,s =3,r =1.
0 0 0 0
(109)
fma1  fma2 fsb2 mrc2 
 '4  t  =1 +2      gmc2 
K  K K K 
 mra3 rsb3 
3    gma3   gsb3   4 b4
 K K 
with the transversality condition gives as
1(T)=2(T)=3(T)=4(T)=0 (102)
Now considering the optimality conditions, the
Hamiltonian function is differentiated with respect to
control variable  resulting in
H
=  (103)
 4
Then a compact way of writing the optimal control  is
*(t)=max(0,4) (104)
Theorem 7.2 An optimal control *U for the system

f+m+s+r Fig 3:- Here, we simulate case 1 where g=1 and (r)= .
(9292) with g=1 ,(r)= that maximizes the
K
objective functional J is characterized by (104).
7.5 g=1,(r)=r2
 '1  t  =1-1  ma1  b1  -2  ma2 +sb 2  ,

 '2  t  =1-1a1 f -2  fa2 +rc 2  e2  -3 ra 3 ,
 '3  t  =-2 b 2 f -3  rb3 +e3  , (105)
 '4  t  =-2 c2 m-3  ma3 +sb3  +4 b4 ,

with the transversality condition gives as
1(T)=2(T)=3(T)=4(T)=0 (106)
Now considering the optimality conditions, the
Hamiltonian function is differentiated with respect to
control variable  resulting in f+m+s+r
Fig 4:- Here, we simulate case 2 where g=1 and
H 2 K
=r   (107)
 4 (r)=.
Then a compact way of writing the optimal control  is

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f+m+s+r
Fig 8: Here, we simulate case 2 where g=1 and
K
(r)=.
Fig 5:- Here, we simulate case 3 where g=1 and (r)=r2.
A B
Figure 6: We take the initial conditions as
f =20,m =20,s =3,r =1 and compare the decay
0 0 0 0
Fig 9:- Here, we simulate case 3 where g=1 and (r)=r2.
rate of both females and males in each case.
Then we simulate the system with larger initial conditions,

For small initial population, cases 1-3 are all effective
f0=90,m0=90,s0=8,r0=6. (110)
to eradicate wild females and males; however, case 3
requires larger  for a certain period which is shown in
Fig.3-5. As for case 1 and case 2, it seems that Introducing
 does not help to eradicate the invasive species, so the
*
optimal control  is almost identically 0. In the Fig.6, we
can clearly see there is a sharper decay in both f and m
under optimal control. The decay rate for case 1 and 3 are
almost the same under the optimal control.
For large initial population, case 1 and case 3 do not

eradicate the whole population no matter how large  is,
which is shown in Fig.7 and Fig.9. We can also conclude
that large  does not help to eradicate the population, and
depending on parameters and initial conditions, the
population could blow up when g=1. However, even with
large initial population, case 2 can always eradicate the
whole population as long as we can provide enough ,
Fig 7:- Here, we simulate case 1 where g=1 and (r)=. which is shown in Fig.8.

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VIII. DISCUSSION AND CONCLUSIONS which we have finite time blow up. The point here is to
show that in the good parameter range of data and
The use of Trojan sex chromosomes is an approach parameters, the asymptotic behavior of the system is a one
for eradicating invasive species that have a XY sex point attractor, that is, for a given set of parameters and
determination system and for which it is feasible to force data where we have global existence, the solutions in long
sex reversal. It was clearly established that extinction is time tend towards a steady state (not necessarily spatially
possible in the supermale dynamical system as a function of uniform in all components). Furthermore, this can be the
the rate  of introduction of supermales (s). The TYC extinction state, depending on the size parameter . Thus
system depends upon parameters that can be deduced from the system can always be driven to the extinction state, via
observations, including the carrying capacity (K), the death the introduced genetically modified organism. This
coefficients (g , g , g ), and the birth coefficients (g , g , validates our control strategy, and asserts that in principle,
1 2 3 2 5
g6), see (7). Further refinement to these parameters should we can always combat invasive species even when used in
the context of bio-terrorism, via our proposed strategy [53,
be made from current field data [62-15]. 54 and 55].
The existence of a bounded absorbing set indicates The analysis of global attractors can be helpful to
that for either eradication or invasion the final state of the estimate times to extinction in complex spatial domains.
population is stable. Via Theorem 6.8 we showed that given We have determined that for Dirichlet boundary conditions
the initial population and values for the parameters, it is on a connected domain there exists an extinction state as a
possible to find an explicit time such that for times greater result of the introduction of s. However, more complicated
than this, an attractor is reached, in which the population is geometries or boundary conditions could have an influence
confined to finite sized sets in H. That is there is a compact in coexistence or extinction. Increasing the level of
subset of the phase space, that attracts all trajectories, in the sophistication of the eradication strategy, the distribution of
dynamical system. Furthermore, knowing the analytical s individuals could be variable in space as opposed to the
form of the bounded absorbing sets, helps guide the constant level that has been studied, i.e. s could be a
exploration of the parameter space. This and related population density dependent function intended to
problems are the subject of current investigation [33, 32]. minimize the introduction of s individuals and therefore
We numerically see that the attractor is a one point minimize costs of implementation. Also under our
attractor, depending on parameter values. Furthermore, proposed strategy, the system can always be driven to the
what is also tried in the numerical experiments, is that the extinction state, at an exponential rate. This is seen
form of  is changed from a constant, to being state numerically as well in Fig1, where we compare the
dependent. trajectories of normal males and females to the function
Also we tried numerical experiments, where we use a e(0.012)t. Clearly, exponential attraction to the extinction
state is seen. It would be interesting to try and rigorously
bad source term such as r2. In this case we compare (88) - prove the existence of an exponential attractor, in this
(91), with g as a logistic term, with the same system when setting.
 is replaced with r2. Surprisingly, this causes a slower
decay in the female species, than if one were to use , see The viability of YY individuals remains an open
question. The supermale model assumes that phenotypes
Fig2. We also tried simulations with r3 and r4. What we are stable after maturation, but this could be problematic
observe is that as the power on the source term increases, for species whose sex determination involves many genes,
the decay rate of the female species gets slower. The point or when there is environmental pressure to feminization or
here is one can always stop the influx of the feminized masculinization. To incorporate this we choose death
supermale, before the actual blow up, if the bad source coefficients so that the supermales die at a faster rate than
actually “sped up" the extinction, but this is not seen to the normal males, as these are not considered as fit as their
happen. It is however worth investigating other source normal counterparts, [34], and so fitness penalty should be
terms, in this context. The optimal control experiments tell exercised. Another potential problem is hybridization with
us that in general, whether the initial population of the compatible species, which would extend the eradication
invasive species is large or small, the classic TYC strategy pressure beyond the initial target; however, this effect
(case 2) is always effective from the perspective of should disappear by the interruption of the influx of s. Also
eradication of the invasive species. The larger the initial it would make for very interesting future work if we could
population of the invasive species, the larger the perhaps place sufficient restrictions on the reaction terms in
introduction of superfemales (s) has to be to achieve question to show via Simon-Lojasiewicz gradient
eradication. Optimal control of the TYC system under inequality techniques [49], to show that convergence to the
various parameter regimes is also an area of current spatially homogenous equilibrium state is guaranteed.
investigations [63] that should be pursued further.
To summarize, we have rigorously shown that
We would like to point out that since g ,1i10 are all introduction of phenotypically manipulated supermales into
i
constants in some of our simulations, the conditions for an established population can lead to local extinction.
global existence via (20)-(23) are not met. That is there can Moreover, this can be done even if the population dynamics
be various range of large initial data, and parameters, for

ISSN No:-2456-2165
of the species involved, is not governed by a logistic type [13]. J. B. Gutierrez and J. L. Teem, A model describing
control term. the effect of sex-reversed (YY) fish in an established
wild population: the use of a Trojan Y chromosome to
ACKNOWLEDGEMENTS cause extinction of an introduced exotic species,
Journal of Theoretical Biology. 241(22) (2006), 333-
The authors Said Kouachi and Rana D. Parshad 341.
gratefully acknowledge Qassim University, represented by [14]. Juan B. Gutierrez, Rana D. Parshad, John L.Teem and
the Deanship of Scientific Research, on the material Monica Hurdal, Analysis of the Trojan Y Chromosome
support for this research under the number ( 1327-cos- model for eradication of exotic species in a dendritic
2016-1-12-I ) during the two academic years 1438-1440 riverine system, Journal of Mathematical Biology.
AH / 2017-2019 AD. Rana D. Parshad and Jingjing Lyu 64(1-2) (2012), 319-340.
also acknowledge valuable partial support from the [15]. Kennedy, Patrick A., et al. Survival and Reproductive
National Science Foundation via awards DMS-1715377 Success of Hatchery YY Male Brook Trout Stocked in
and DMS-1839993. Idaho Streams, Transactions of the American
Fisheries Society 147(3): 419-430, 2018.
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Volume 4, Issue 4, April – 2019 International Journal of Innovative Science and Research Technology

Global Dynamics of a PDE Model for Eradication of

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II. THE MATHEMATICAL MODEL where r (t ) is a bounded function on bounded subsets of

Definition. 3.1. We say that

Our aim is to construct polynomial Lyapunov functionals

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last method, implies in several cases the global existence in and

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L'p (t ) Remark 8 Typically, in order to make dissipative

L2(). Similar estimates are made in [48]. e2t1 t

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This leads us to state the following Lemma. p

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Similar estimates can be derived for the m,s components.

 C,   t6* , * ( 2 1) * (  2 1)

We will next make a uniform in time estimate for ,

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In this section we prove the existence of a compact

{tn}, S(tn)f0,n possesses a convergent subsequence

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fnf strongly in H2().

This proves the Lemma.

Theorem 6.8. Consider the reaction diffusion system

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U={| measurable, 0<, t[0,T], T}. (94)

7.2 Optimal Control  '4  t  =-2 c2 m-3  ma3 +sb3  +4 b4 ,

f Now considering the optimality conditions, the

We will use optimal control theory to illustrate which

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with the transversality condition gives as

Theorem 7.2 An optimal control *U for the system

 '1  t  =1-1  ma1  b1  -2  ma2 +sb 2  ,

 '4  t  =-2 c2 m-3  ma3 +sb3  +4 b4 ,

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Then we simulate the system with larger initial conditions,

For large initial population, case 1 and case 3 do not

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