On the Well Posedness and Refined Estimates for the Global Attractor of the TYC Model

Volume 2010, Article ID 405816,29pages doi:10.1155/2010/405816

Research Article

On the Well Posedness and Refined Estimates for the Global Attractor of the TYC Model

Rana D. Parshad

and Juan B. Gutierrez

1Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13676, USA

2Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, USA

Correspondence should be addressed to Rana D. Parshad,[email protected] Received 14 July 2010; Accepted 2 November 2010

Academic Editor: Sandro Salsa

Copyrightq2010 R. D. Parshad and J. B. Gutierrez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Trojan Y Chromosome strategyTYC is a theoretical method for eradication of invasive species. It requires constant introduction of artificial individuals into a target population, causing a shift in the sex ratio that ultimately leads to local extinction. In this work we demonstrate the existence of a unique weak solution to the infinite dimensional TYC system. Furthermore, we obtain improved estimates on the upper bounds for the Hausdorffand fractal dimensions of the global attractor of the TYC system, via the use of weighted Sobolev spaces. These results confirm that the TYC eradication strategy is a sound theoretical method of eradication of invasive species in a spatial setting. It also provides a solid ground for experiments in silico and validates the use of the TYC strategy in vivo.

1. Introduction

An exotic species is a species that resides outside its native habitat. When it causes some sort of measurable damage, it is often referred to as invasive. The recent globalization process has expedited the pace at which exotic species are introduced into new environments. Once established, these species can be extremely difficult to manage and almost impossible to eradicate1,2. Studies have indicated that the losses caused by invasive species could be as much as $120 billion/year by 20043. The effect of these invaders is thus devastating4.

Current approaches for controlling exotic fish species are limited to general chemical control methods applied to small water bodies and/or small isolated populations that kill native fish in addition to the target fish5. For example, the piscicide Rotenone has been used to eradicate exotic fish, but at the expense of killing all the endogenous fish, making it necessary to restock native fish from other sources1,2.

A genetic strategy to cause extinction of invasive species was proposed by Gutierrez and Teem6. This strategy is relevant to species amenable to sex reversal and with an XY sex- determination system, in which males are the heterogametic sexcarrying one X chromosome

and one Y chromosome, XY and females are the homogametic sex carrying two X chromosomes, XX. The strategy relies on the fact that variations in the sex chromosome number can be produced through genetic manipulation, for example, a normal and fertile male bearing two Y chromosomessupermale, YY 7–10. Also hormone treatments can be used to reverse the sex, resulting in a feminized YY supermale5,11,12.

The eradication strategy requires adding a sex-reversed “Trojan” female individual bearing two Y chromosomes, that is, feminized supermales r, at a constant rate μ to a target population of an invasive species, containing normal females and males denoted as f and m, respectively. Matings involving the introduced r generate a disproportionate number of males over time. The higher incidence of males decrease the female to male ratio.

Ultimately, the number offdecline to zero, causing local extinction. This theoretical method of eradication is known as Trojan Y ChromosomeTYCstrategy.

The original model considered by Gutierrez and Teem was an ODE model. Spatial spread is ubiquitous in aquatic settings and was thus considered by Gutierrez et al. 13, resulting in a PDE model. In14, we considered the PDE model and showed the existence of a global attractor for the system, which isH²Ωregular, attracting orbits uniformly in the L²Ωmetric. We showed that this attractor supports a state, in which the female population is driven to zero, thus resulting in local extinction. Recall the TYC model with spatial spread takes the following form14:

∂f

∂t DΔf1

2fmβL−δf, f

∂Ω 0, 1.1

∂m

∂t DΔm

1 2fm1

2rmfs

βL−δm, m|_∂Ω 0, 1.2

∂s

∂t DΔs 1

2rmrs

βL−δs, s|_∂Ω 0, 1.3

∂r

∂t DΔrμ−δr, r|_∂Ω 0. 1.4

Here,Ω⊂R³is a bounded domain. Also

L 1−

fmrs K

, 1.5

whereKis the carrying capacity of the ecosystem,Dis a diffusivity coefficient,δis a birth coefficienti.e., what proportion of encounters between males and females result in progeny, and δ is a death coefficienti.e., what proportion of the population is dying at any given moment. We assume initial data is positive and inL²Ω. At the outset we would like to point out that the difficulty in analyzing1.1–1.4lies in the nonlinear termsLfm,L1/2fm 1/2rmfsandL1/2rmrs. See15for a PDE dealing with similar nonlinearities, albeit in the setting of a fluid-saturated porous medium. We will also assume positivity of solutions as negativef, m, r, sdo not make sense in the biological context. We also provide a rigoros proof to this end.

In the current paper we will show that the TYC model,1.1–1.4, possesses a unique weak solutionf, m, r, s. By this we mean that there existf, m, r, ssuch that the following is satisfied in the distributional sense:

d dt

f, v D

∇f,∇v δ

f, v 1

2fmβL, v

, d

dtm, vD∇m,∇vδm, v 1 2fm 1

2rmfs

βL, v

, d

dtr, vD∇s,∇vδs, v 1

2rmrs

βL, v

, d

dts, vD∇s,∇vδs, v μ, v

.

1.6

Here, · is the standard inner product in L²Ω. Furthermore the above hold for all v ∈ H₀¹Ω. Our main result is summarized in the following theorem.

Theorem 1.1. Consider the Trojan Y Chromosome model,1.1–1.4. There exists a unique weak solutionf, m, r, sto the system for positive initial data inL²Ω, such that

f, m, r, s

∈C

0, T;L²Ω

∩L^∞

0, T;L²Ω

∩L²

0, T;H₀¹Ω , ∂f

∂t,∂m

∂t ,∂r

∂t,∂s

∂t

∈L²

0, T;H⁻¹Ω ,

1.7

for allT >0. Furthermore,f, m, r, sare continuous with respect to initial data.

Our strategy to prove the above is as follows: we first derive a priori estimates for thef, m, r, svariables. We then show existence of a solution to1.1. Note, showing existence of a solution to1.1requires a priori estimates onm, r, salso. The key here isLemma 4.1 which enables convergence of the nonlinear term Lfm. Next we show uniqueness of the solution to 1.1. The procedure to show existence and uniqueness of solutions to 1.2–

1.4follow similarly. We then consider the question of sharpening the upper bounds on the Hausdorffand fractal dimension of the global attractor for the system, derived in14. This constitutes our second main result,Theorem 7.2. Lastly, we offer some concluding remarks.

In all estimates made hence, forth,Cis a generic constant that can change in its value from line to line and sometimes within the same line if so required.

2. A Bound in L

Ω

The biology of the system dictates that the solutions are bounded in the supremum norm by the carrying capacity. We now provide a proof via a maximum principle argument.

Lemma 2.1. Consider the Trojan Y Chromosome model,1.1–1.4. The solutionsf, m, r, sof the system are bounded as follows:

f

∞≤K,

|m|_∞≤K,

|s|_∞≤K,

|r|_∞≤K.

2.1

Proof. The proof relies heavily on the form of the nonlinearity in the system. We concentrate on the nonlinear term in1.1,

F

f, m, r, s fm

1−fmrs K

. 2.2

The analysis for the other terms is similar. As is biologically viable, we assumesf,m,r, and sare always positive, thus, we have

f >0, m >0, r >0, s >0. 2.3

Assuming positive initial data,f₀ >0,m₀ >0,r₀ >0, ands₀ > 0, the solution at later times remains positive. In order to prove this let us assume the contrary, that isf0>0,m0 >0,r0>0, ands₀>0, but sayfcan become negative at a later time. Consider an interior minimum point in the parabolic cyliderΩ×0, T, that is somex^∗, t^∗, such thatf attains a minimum there, and thatfx^∗, t^∗ < 0,mx^∗, t^∗ < 0,rx^∗, t^∗ < 0, andsx^∗, t^∗ < 0. Under this setting, from standard calculus, we have

∂f

∂tx^∗, t^∗ 0, Δfx^∗, t^∗≥0, 2.4

furthermore,

−δfx^∗, t^∗>0,

βfx^∗, t^∗mx^∗, t^∗

1−fx^∗, t^∗ mx^∗, t^∗ rx^∗, t^∗ sx^∗, t^∗ K

>0.

2.5

Thus from1.1, we have

∂f

∂tx^∗, t^∗ 0

Δfx^∗, t^∗−δfx^∗, t^∗ βfx^∗, t^∗mx^∗, t^∗

1−fx^∗, t^∗ mx^∗, t^∗ rx^∗, t^∗ sx^∗, t^∗ K

>000 0.

2.6

This is clearly a contradiction. Thus even at an interior minimumf >0, hencef >0 everywhere else. The same argument can be applied on the equations describing them,r, andsvariables. Actually the equation forris exactly solvable and is seen to be positive. Thus our assumption via2.3is feasible. Thus we proceed with our proof via maximum principle.

Despite not biologically viable, assume for purposes of analysis that

f≥K≥1, m≥K≥1. 2.7

We now define the positive and negative parts off−Kas f−K

x

⎧⎨

⎩

f−K, f > K, 0, otherwise, f−K

−x

⎧⎨

⎩

f−K, f < K, 0, otherwise.

2.8

We now multiply1.1byf−Kxand integrate by parts to yield d

dtf−K

²

2∇ f−K

²

2δf−K

²

2 ≤

ΩF

f, m, r, s f−K

xdx. 2.9 Whenf < Kthe right-hand side is zero. Whenf > K, assumingf ≥Kwhere >0, and m > kvia2.3, we have

ΩF

f, m, r, s f−K

xdx

Ωfm

1−fmrs K

f−K

xdx

≤

ΩKK

1−fmrs K

dx

≤

ΩKK

1−2K2δ K

dx

≤ |Ω|K²

−1−2δ K

≤ |Ω|K²

−2δ K

≤0.

2.10

Hence, via Poincar´e’s Inequality, we obtain d

dtf−K

²

2 Cδf−K

²

2≤0. 2.11

Application of Gronwall’s Lemma now yields f−K

²

2≤e^−Cδtf₀−K

²

2. 2.12

We can now considert → ∞to yield

f−K

0. 2.13

The same argument on the negative part offyields, f−K

− 0. 2.14

Since the positive and negative parts offcan be no more thanK, we obtain f

∞≤K. 2.15

The same technique works onm, s, andrand is trivially seen to be bounded from the form of1.4.

3. A Priori Estimates

In order to prove the well posedness we follow the standard approach of projecting onto a finite dimensional subspace. This reduces the PDE to a finite dimensional system of ODE’s.

It is on this truncated system that we make a priori estimates. Essentially The truncation for ftakes the form

fnt ⁿ

j 1

fnjtwj. 3.1

Here w_j are the eigenfunctions of the negative Laplacian, so −Δwi λiwi. A similar truncation can be performed for m, r and s. Thus, essentially the following holds for all 1≤j≤n,

∂fn

∂t DΔfnPn

F

fn, mn, rn, sn

−δfn, 3.2

fn0 Pn

f0

. 3.3

HerePnis the projection onto the space of the firstneigenvectors. Note in general f_n, P_n

F

f_n

P_n f_n

, F

f_n f_n, F

f_n

. 3.4

We multiply3.2byf_nand integrate by parts overΩ. We thus obtain 1

2 df_n²

2

dt −D∇fn²

2β

2 _Ωm_nf_n²dx−

Ωm_nf_n²f_nm_nr_ns_n

K dx

−δf_n²

2. 3.5

Via the positivity offn,mn,rn,sn, andKit follows that

Ωm_nf_n²f_nm_nr_ns_n

K dx≥

Ωm_nf_n²f_n

Kdx. 3.6

This estimate is used in3.5to yield 1

2 dfn²

2

dt D∇fn²

2δfn²

2 β

2K

Ωmnf_n³dx≤ β 2

Ωmnf_n²dx. 3.7 We now use Young’s Inequality to obtain

1 2

df_n²

2

dt D∇fn²

2δf_n²

2 β

2K

Ωm_nf_n³dx≤ β 2K

Ωm_nf_n³dxβK² 2

Ωm_ndx. 3.8 Using

|mn|_∞≤ |m|_∞≤K, 3.9

we obtain the following 1 2

dfn²

2

dt D∇fn²

2δfn²

2 ≤ βK³

2 |Ω|. 3.10

The use of Poincar´e’s Inequality yields dfn²

2

dt CDδfn²

2≤βK³|Ω|. 3.11

Now, we can apply Gronwall’s Lemma to yield fnt²

2≤e^−CDδtf0²

2 βK³|Ω|

CDδ ≤C, ∀t≥0. 3.12

On the other hand we can integrate3.10from 0 toTto obtain 1

2f_nT²

2D _T

0

∇fn²

2dtδ _T

0

f_n²

2dt≤ _T

0

βK³|Ω|dtf_n0²

2. 3.13

This immediately yields _T

0

∇fn²

2dt≤ _T

0

βK³|Ω|dtfn0²

2

≤ _T

0

βK³|Ω|dtf0²

2

≤C.

3.14

Thus, via3.12and3.14, we obtain fn∈L^∞

0, T;L²Ω

, 3.15

f_n∈L²

0, T;H₀¹Ω

. 3.16

3.2. Estimate for the Time Derivative offn

We multiply3.2by aw∈H₀¹Ωto yield ∂f_n

∂t , w

−D

∇fn,∇w

F

f_n, m_n, r_n, s_n

, P_nw

−δ f_n, w

. 3.17

We estimate the nonlinear term as follows:

F f_n

, P_nw

Ωm_nf_n

1−f_nm_nr_ns_n K

P_nwdx

≤

Ωm_nf_nP_nwdx

≤ |m_n|_∞

Ωf_nP_nwdx

≤Kfn

4|Pnw|_4/3

≤Cf_n

4|w|_H¹

0.

3.18

This follows via the compact embedding ofH₀¹Ω→L^4/3Ω. Thus, we have ∂fn

∂t ²

H⁻¹Ω≤f_n²

4. 3.19

Integrating both sides of the above in the time interval0, Tyields _T

0

∂fn

∂t ²

H⁻¹Ωdt≤ _T

0

fn²

4dt≤C _T

0

∇fn²

2dt≤C. 3.20

This follows from the derived estimate via3.16and the compact embedding ofH₀¹Ω → L⁴Ω. Thus, we obtain

∂f_n

∂t ∈L²

0, T;H⁻¹Ω

. 3.21

We can now via3.15and3.16extract a subsequencefnjsuch that

f_nj f^∗ inL^∞

0, T;L²Ω , f_nj f inL²

0, T;H₀¹Ω , f_nj −→f inL²

0, T;L²Ω .

3.22

The convergence in the last equation follows via the compact embedding ofH₀¹Ω→L²Ω.

3.3. A Priori Estimates form,r, and s

The a priori estimates form,randsare very similar to the estimates forf. We omit the details here and present the results.

The truncation formsatisfies the following a priori estimates:

m_n∈L^∞

0, T;L²Ω , m_n∈L²

0, T;H₀¹Ω ,

∂sn

∂t ∈L²

0, T;H⁻¹Ω .

3.23

We can now extract a subsequencemnj such that

m_nj s^∗ inL^∞

0, T;L²Ω , m_nj s inL²

0, T;H₀¹Ω , m_nj −→s inL²

0, T;L²Ω .

3.24

The last inequality follows via the compact embedding of

H₀¹Ω→L²Ω. 3.25

The truncation forssatisfies the following a priori estimates:

s_n∈L^∞

0, T;L²Ω , s_n∈L²

0, T;H₀¹Ω ,

∂sn

∂t ∈L²

0, T;H⁻¹Ω .

3.26

We can now extract a subsequencesnjsuch that

snj

s∗ inL^∞

0, T;L²Ω , snj s inL²

0, T;H₀¹Ω , snj −→s inL²

0, T;L²Ω .

3.27

The last inequality follows via the compact embedding of

H₀¹Ω→L²Ω. 3.28

The truncation forrsatisfies the following a priori estimates:

rn∈L^∞

0, T;L²Ω , rn∈L²

0, T;H₀¹Ω ,

∂r_n

∂t ∈L²

0, T;H⁻¹Ω .

3.29

We can now extract a subsequencer_njsuch that

r_nj r^∗ inL^∞

0, T;L²Ω , r_nj r inL²

0, T;H₀¹Ω , r_nj−→r inL²

0, T;L²Ω .

3.30

The last inequality follows via the compact embedding of

H₀¹Ω→L²Ω. 3.31

4. Existence of Solution

We recast1.1in the following form:

∂f

∂t DΔfF

f, r, m, s

−δf, f

∂Ω 0. 4.1

Here,

F

f, m, r, s β 2

1−fmrs K

fm. 4.2

Note that the key element in proving the existence will be to show convergence of the nonlinear termFfnj, mnj, rnj, snjtoFf, m, r, s. To this end we state the following lemma, Lemma 4.1. Consider the non linear termsFf1, m₁, r₁, s₁andFf2, m₂, r₂, s₂as defined via4.2.

The following estimate for their difference holds F

f1, m1, r1, s1

−F

f2, m2, r2, s2

2≤Cf1−f2

2|m1−m2|₂|s1−s2|₂|r1−r2|₂ . 4.3

Proof. Via4.2, we have that F

f₁, m₂, r₂, s₂

−F

f₂, m₂, r₂, s₂ f₁m₁−f₂m₂−

f₁²m₁−f₂²m₂

−

m²₁f₁−m²₂f₂ f₁m₁r₁−f₂m₂r₂f₁m₁s₁−f₂m₂s₂

f₁m1−m₂ m₂ f₁−f₂

−

f₁²m1−m₂ m₂

f₁²−f₂²

− m²₁

f₁−f₂ f₂

m²₁−m²₂

m₁r₁ f₁−f₂

f₂m₂r1−r₂ f₂r₁m1−m₂ m₁s₁

f₁−f₂

f₂m₂s1−s₂ f₂s₁m1−m₂.

4.4

We supress the dependence of the right-hand side on the constantβ/2 for convenience.

This follows from standard algebraic manipulation. Application of Holder’s and Minkowski’s inequalities yield

F

f₁, m₂, r₂, s₂

−F

f₂, m₂, r₂, s₂

2

≤f₁

∞|m1−m₂|₂|m2|_∞f₁−f₂

2f₁²

∞|m1−m₂|₂ |m2|_∞f₁f₂

∞f₁−f₂

2|m1|²_∞f₁−f₂

2

f₂

∞|m1m₂|_∞|m1−m₂|₂ |m1|_∞|r1|_∞f1−f2

2f2

∞|m2|_∞|r1−r2|₂|r1|_∞f2

∞|m1−m2|₂ |s1|_∞|m1|_∞f₁−f₂

2f₂

∞|m2|_∞|s1−s₂|₂|s1|_∞f₂

∞|m1−m₂|₂

≤Cf₁−f₂

2|m1−m₂|₂|s1−s₂|₂|r1−r₂|₂ .

4.5

4.2. Passage to Weak Limit

As, we have made the a priori estimates on the truncations, we will attempt to pass to the weak limit, as is the standard practice. We will focus on1.1. Recall via Galerkin truncation we are seeking an approximate solution of the form

fnt ⁿ

j 1

fnjtwj, 4.6

such that, for each 1≤j≤n, and for allφ∈C^∞₀ 0, T, the following holds:

df_n dt, φw_j

D

∇fnj,∇wjφt δ

f_nj, φtwj

F

f_nj , φw_j

, 4.7

f_n0 P_n f₀

. 4.8

Here and henceforth we assume Ffnj P_nFfnj, whereP_n is the projection operator onto the firstneigenvectors. Upon passage to the weak limit of4.7, we will have obtained

df dt, w_j

D

∇f,∇wj

δ

f, w_j F

f , w_j

. 4.9

This will imply the existence of a weak solutionfto1.1. We proceed as follows. Consider a φ∈C^∞₀ 0, T. We multiply4.7byφtand integrate by parts in time to yield

− _T

0

f_nj, φtwj

dt −D _T

0

∇fnj,∇wjφt dt

_T

0

F

f_nj , φtwj

dt

−δ _T

0

fnj, φtwj

dt.

4.10

We will first show convergence of the nonlinear term. This is stated via the following lemma.

Lemma 4.2. Consider the nonlinear termFf, m, r, sas defined via4.2. The following convergence result holds:

jlim→ ∞

_T

0

ΩF

f_nj, m_nj, s_nj, r_nj

φtwjdxdt _T

0

ΩF

f, m, s, r

φtwjdxdt, 4.11

for allφ∈C^∞₀ 0, T. Proof. Consider

lim

j→ ∞

_T

0

ΩF fnj

φtwjdxdt− _T

0

ΩF f

φtwjdxdt

≤C _T

0

Ω

F

fnj

, φwj

− F

f

, φwj²dxdt

≤Cφ

∞w_j

∞

_T

0

Ω

F f

−F

f_nj²dxdt

≤C _T

0

f−f_nj²

2m−m_nj²

2r−r_nj²

2s−s_nj²

2

dt

≤C f−f_nj

L²0,T;L²m−m_nj

L²0,T;L²s−s_nj

L²0,T;L²

C r−rnj

L²0,T;L²

≤C0000 0.

4.12

This follows viaLemma 4.1and because, we have demonstrated fnj −→f inL²

0, T;L²Ω , m_nj −→m inL²

0, T;L²Ω , snj −→s inL²

0, T;L²Ω , r_nj−→r inL²

0, T;L²Ω .

4.13

Thus the convergence of the nonlinear term has been established. Now, taking the limit asj → ∞in4.10, we obtain

jlim→ ∞

_T

0

f_nj, φtwj

dtD _T

0

∇fnj,∇wjφt dt

δ _T

0

f_nj, φw_j dt−

_T

0

F

f_nj , φw_j

dt _T

0

f, φtwj

dtD _T

0

∇f,∇wjφ dt

δ _T

0

f, φw_j dt−

_T

0

F f

, φw_j dt 0.

4.14

The last term on the right-hand side can be bounded as follows _T

0

F f

, φwj

dt≤Cφ

∞

_T

0

Ω

f²wjdt

≤Cφ_∞w_j

2f

L²0,T;L⁴Ω

≤Cφ

∞w_j

H₀¹Ωf

L²0,T;H₀¹Ω

≤Cw_j

H¹₀Ω.

4.15

This follows by the compact embedding ofH₀¹Ω → L⁴Ω → L²Ω. This implies that, we have continuity with respect tow_j. Thus, we obtain that for anyv ∈ H₀¹Ω the following holds

− _T

0

f, φtv dtD

_T

0

∇f,∇vφt dtδ

_T

0

f, φtv dt

_T

0

F f

, φtv

dt. 4.16

This yields the existence of anfsuch that the following is true in a distributional sense d

dt f, v

D

∇f,∇v δ

f, v F

f , v

, ∀v∈H₀¹Ω. 4.17

In other words there exists a weak solutionfto1.1. Since f ∈L^∞

0, T;L²Ω

∩L²

0, T;H₀¹Ω ,

∂f

∂t ∈L²

0, T;H⁻¹Ω ,

4.18

it follows via standard PDE theory, see16,17, that f∈C

0, T;L²Ω

. 4.19

This establishes that the solution belongs to the requisite functional spaces.

4.3. Continuity with Respect to Initial Data and Uniqueness of Solutions We now show continuity with respect to initial data of the solution via the following lemma.

Lemma 4.3. Consider the Trojan Y Chromosome model. For positive initial data inL²Ω, any weak solutionf, m, s, rof the Trojan Y Chromosome model is continuous with respect to initial data, that is,

f0 f0, m0 m0, s0 s0, r0 r0. 4.20

Proof. We will show the details forf, and the other variables follow suit accordingly. We take a test functionφ∈C¹0, Tsuch that

φ0 1, φT 0. 4.21

With this choice ofφtin4.17, we integrate the first term twice by parts to yield

− _T

0

f, φtv dtD

_T

0

∇f,∇vφt dtδ

_T

0

f, φtv dt f0, v

D _T

0

∇f,∇vφt dtδ

_T

0

f, φtv dt.

4.22

Note that the truncation satisfies _T

0

f_nj, φtv dtD

_T

0

∇fnj,∇vφt dtδ

_T

0

f_nj, φtv dt

fnj0, v D

_T

0

∇fnj,∇vφt dtδ

_T

0

fnj, φtv dt.

4.23

Thus, taking the limit asj → ∞in4.28just as done earlier yields

− _T

0

f, φtv dtD

_T

0

∇f,∇vφt dtδ

_T

0

f, φtv dt f₀, v

D _T

0

∇f,∇vφt dtδ

_T

0

f, φtv dt.

4.24

Thus, we obtain

f0, v f0, v

, ∀v∈H₀¹Ω. 4.25

This yields

f0 f₀, 4.26

as is required.

We now state the uniqueness result via the following lemma.

Lemma 4.4. Consider the Trojan Y Chromosome model. For positive initial data inL²Ωany weak solutionf, m, s, rof the Trojan Y Chromosome model is unique.

Proof. We work out the case for thefvariable, uniqueness for the others follow similarly. We consider the difference of two solutionsf1andf2to1.1. We denote

w f₁−f₂, 4.27

andwsatisfies the following equation:

dw

dt −DΔwδw F f₁

−F f₂

, 4.28

w0 f₁0−f₂0 0. 4.29

We can multiply4.28bywand integrate by parts overΩto yield d|w|²₂

dt D|∇w|²₂δ|w|²₂

Ω

F f₁

−F f₂

wdx. 4.30

Via the uniformL²estimates onm, r, s, see14, andLemma 4.1, we obtain d|w|²₂

dt D|∇w|²₂δ|w|²₂≤Cf₁−f₂

2|w|₂≤CK|w|²₂. 4.31 This yields

d|w|²₂

dt D|∇w|²₂δ|w|²₂−CK|w|²₂≤0. 4.32 Now using Poincar´e’s Inequality, we obtain,

d|w|²₂

dt Dδ−C|w|²₂≤0. 4.33

The use of Gronwall’s Lemma yields that for anyt >0 the following estimate holds:

|wt|²₂≤e^−Dδ−Ct|w0|²₂≤0. 4.34

Equation4.17in conjunction withLemma 4.4yieldsTheorem 1.1.

5. Weighted Sobolev Spaces

The purpose of this section is to introduce weighted Sobolev spaces into the framework of our present problem. We will show thatr given by1.4, remains bounded in the norms of these spaces. This will enable us to state a theorem about the existence of weak solution in the weighted spaces. This in turn will entail making refined estimates on the dimension of the global attractor for TYC system, when the phase space is a weighted Sobolev space. This will be achieved via the elegant technique of projecting the trace operator onto a weighted Sobolev space. We first make certain requisite definitions.

Definition 5.1. The weighted Sobolev spaceW_ωx^k,p , with weight functionωx, is defined to be the space consisting of all functionsusuch that

⎛

⎝

|α|≤k

Ω|D^αu|^pωxdx

⎞

⎠

1/p

<∞. 5.1

Remark 5.2. Here,D^α is theαth weak derivative ofu. In particular, we are interested in the following spaces for our application:

L²_ωΩ

u:

Ωωx|u|²dx 1/2

<∞

,

H_0,ω¹ Ω

u:|u|_2,ω|∇u|_2,ω<∞ .

5.2

Also, we denote

Ωωx|u|²dx^1/2 |u|2,ω. We defineH_ω⁻¹Ωto be the dual ofH_0,ω¹ Ω.

5.1. Estimates forr in Weighted Sobolev Spaces Recall the equation forr

∂r

∂t DΔr−δrμ, r|_∂Ω 0. 5.3

We chooseωx e^μx,μ >0, multiply5.3byre^μx, and integrate by parts overΩto yield

1 2

d dt

Ω|r|²e^μxdx −D

Ω|∇r|²e^μxdx−D

Ω∇r· ∇re^μxdx−δ

Ω|r|²e^μxdx μ

Ωre^μxdx

≤ −D

Ω|∇r|²e^μxdx D 2

Ω|∇r|²e^μxdxμ² 2

Ω|r|²e^μxdx

−δ

Ω|r|²e^μxdxμ

Ωre^μxdx

≤ −D 2

Ω|∇r|²e^μxdx−δ

Ω|r|²e^μxdxC μ²K²

2 μK |Ω|.

5.4

These follow via integration by parts, the estimate |r|_∞ ≤ K, and the Cauchy-Schwartz inequality. Thus, we obtain

1 2

d

dt|r|²_2,ωD

2|∇r|²_2,ωδ|r|²_2,ω≤C μ²K²

2 μK |Ω|. 5.5

The use of Poincaire’s Inequality gives us 1

2 d|r|²_2,ω

dt

D 2 δ

|∇r|²_2,ω≤C μ²K²

2 μK |Ω|. 5.6

Now, we can apply the Gronwall Lemma to yield

|rt|²_2,ω≤e^−CDδt|r0|²_2,ωμ²K²/2μK

CDδ , ∀t≥0. 5.7

On the other hand we can integrate5.5from 0 toT to obtain 1

2|rT|²_2,ωD 2

_T

0

|∇r|²_2,ωdtδ _T

0

|r|²_2,ωdt≤ _T

0

μ²K²

2 μK |Ω|dt. 5.8

This immediately yields _T

0

|∇r|²_2,ωdt≤ _T

0

μ²K²

2 μK |Ω|dt. 5.9

Thus, via5.7and5.9, we have that

|r|_L^∞_0,T;L²_ωΩ≤C <∞, 5.10

|r|_L20,T;H¹_0,ωΩ≤C <∞. 5.11

5.2. Estimate for the Time Derivative ofr in Weighted Sobolev Space We multiply5.3by aw∈H_0,ω¹ Ωto yield

∂r

∂t, w

2,ω

−D∇r,∇w_2,ω−δr, w_2,ω w, μ

2,ω, ∂r

∂t

Hω⁻¹Ω≤μ|w|_2,ω.

5.12

Integrating both sides in time from 0 toTyields _T

0

∂r

∂t ²

Hω⁻¹Ωdt≤μ _T

0

|w|²_2,ω

dt. 5.13

Because of the estimate via5.11and the embedding of

H_0,ω¹ Ω→L²_ωΩ, 5.14

we have

∂r

∂t ∈L²

0, T;H_ω⁻¹Ω

< C <∞. 5.15

Thus it follows via the standard functional analysis theory, see16, that r∈C

0, T;L²_ωΩ

. 5.16

These estimates show that r remains bounded in the appropriate weighted spaces introduced earlier and thus enables us to state the following theorem.

Theorem 5.3. Consider1.4in the TYC system. For positiver₀∈L²_ωΩ, there exists a unique weak solutionrto the system with

r∈C

0, T;L²_ωΩ

∩L^∞

0, T;L²_ωΩ

∩L²

0, T;H_0,ω¹ Ω ,

∂r

∂t ∈L²

0, T;H_ω⁻¹Ω .

5.17

Furthermore, the solutions are continuous with respect to initial data.

The uniqueness and convergence result by mimicking the method of proof for Theorem 1.1.

6. Existence of Global Attractor in Weighted Sobolev Space

We recall the following spaces from14, as the natural phase space for our problem:

H L²Ω×L²Ω×L²Ω×L²Ω, Y H₀¹Ω×H₀¹Ω×H₀¹Ω×H₀¹Ω, X H²Ω×H²Ω×H²Ω×H²Ω.

6.1

We next state the following definition.

Definition 6.1. Consider a semigroupStacting on a phase spaceM, then the global attractor A ⊂Mfor this semigroup is an object that satisfies

iAis compact inM.

iiAis invariant, that is,StA A, t≥0.

iiiIfBis bounded inM, then

distMStB,A−→0, t−→ ∞. 6.2

We showed in 14 that there exists a H, X global attractor for the TYC system.

That is an attractor that is compactX, and attracts bounded subsets inHin theXtopology.

Furthermore we showed this attractor had finite fractal and Hausdorffdimension. Our goal now is to improve these estimates, on a somewhat different attractor, via the technique of weighted Sobolev spaces. To this end we define

H! L²Ω×L²Ω×L²Ω×L²_ωΩ,

"

Y H₀¹Ω×H₀¹Ω×H₀¹Ω×H_0,ω¹ Ω. 6.3

Hereωis the weight as introduced earlier. We will first demonstrate the existence of aH,! H! attractor for the TYC system. We will then provide estimates for its Hausdorffand fractal dimensions. The following proposition is stated next.

Proposition 6.2. Consider the TYC system,1.1–1.4. There exists aH,!H! global attractorA"

for the this system which is compact and invariant inH and attracts bounded subsets of" H!in theH! metric.

The proof follows readily by applying the techniques of14to the weighted spaces in question. Recall that there are two essential ingredients to show the existence of a global attractor. The existence of a bounded absorbing set and the asymptotic compactness of the semigroup, see18. Thus we will just focus onr, as the proof for the other variables is the

same as in14. We will prove the above proposition via two lemmas. The first of these is stated next.

Lemma 6.3. Consider the equation forr,1.4, in the TYC system. Forr₀ ∈ L²_ωΩthere exists a bounded absorbing set forrinL²_ωΩ.

Proof. Recall, via5.7, we have

|rt|²_2,ω≤e^−CDδt|r0|²_2,ωμ²K²/2μK

CDδ , ∀t≥0, 6.4

Now consider a timet₁such that

t1 max

⎛

⎜⎝0,ln

|r0|²_2,ω CDδ

⎞

⎟⎠. 6.5

It follows that for any timet > t₁the following uniform estimate holds

|rt|²_2,ω≤1μ²K²/2μK

CDδ ≤C. 6.6

This gives us a bounded absorbing set forrinL²_ωΩ.

We next state the following lemma.

Lemma 6.4. The semigroupStfor the TYC system,1.1–1.4, is asymptotically compact inH.! Proof. We again demonstrate the proof forr. Multiply5.3by−Δre^μxand integrate by parts overΩto yield

1 2

d dt

Ω|∇r|²e^μxdx≤ −D

Ω|Δr|²e^μxdx−δ

Ω∇r· ∇re^μxdx μ

Ω

∇r∂r

∂te^μx dx.

6.7

Now Poincaire’s Inequality along with Cauchy-Schwartz imply that 1

2 d dt

Ω|∇r|²e^μxdxCDδ

Ω|∇r|²e^μxdx≤C

|∇r|²₂ ∂r

∂t ²

2

. 6.8

However directly from1.4and the compact Sobolev embedding of

H²Ω→H₀¹Ω. 6.9