A Regular Gonosomal Evolution Operator with uncountable set of fixed points
AA REGULAR GONOSOMAL EVOLUTION OPERATOR WITHUNCOUNTABLE SET OF FIXED POINTS
A.T. ABSALAMOV AND U.A. ROZIKOV
Abstract.
In this paper we study dynamical systems generated by a gonosomal evolutionoperator of a bisexual population. We find explicitly all (uncountable set) of fixed points ofthe operator. It is shown that each fixed point has eigenvalues less or equal to 1. Moreover,we show that each trajectory converges to a fixed point, i.e. the operator is reqular. Thereare uncountable family of invariant sets each of which consisting unique fixed point. Thusthere is one-to-one correspondence between such invariant sets and the set of fixed points.Any trajectory started at a point of the invariant set converges to the corresponding fixedpoint. Introduction
Population dynamics theory is important to a proper understanding of living populationsat all levels. This is a well developed branch of mathematical biology, which has a historyof more than two hundred years.The book [3] contains a short history of applications of mathematics to solving vari-ous problems in population dynamics. For background and motivations of the theory ofpopulation dynamics see [1]-[16].In this paper we consider a bisexual population which consists females partitioned intotypes indexed by { , , . . . , n } and the males partitioned into types indexed by { , , . . . , ν } (see [8], [10], [14] for details).Let γ ( f ) ik,j and γ ( m ) ik,l be inheritance coefficients defined as the probability that a femaleoffspring is type j and, respectively, that a male offspring is of type l , when the parentalpair is ik ( i, j = 1 , . . . , n ; and k, l = 1 , . . . , ν ). These quantities satisfy the following γ ( f ) ik,j ≥ , γ ( m ) ik,l ≥ , (cid:80) nj =1 γ ( f ) ik,j + (cid:80) νl =1 γ ( m ) ik,l = 1 , for all i, k, j, l. (1.1)Define ( n + ν − − dimensional simplex: S n + ν − = s = ( x , . . . , x n , y , . . . , y ν ) ∈ R n + ν : x i ≥ , y j ≥ , n (cid:88) i =1 x i + ν (cid:88) j =1 y j = 1 . Mathematics Subject Classification.
Key words and phrases.
Dynamical systems; fixed point; invariant set, limit point. a r X i v : . [ m a t h . D S ] F e b A.T. ABSALAMOV AND U.A. ROZIKOV
Denote O = (cid:8) s ∈ S n + ν − : ( x , . . . , x n ) = (0 , . . . ,
0) or ( y , . . . , y ν ) = (0 , . . . , (cid:9) . S n,ν = S n + ν − \ O . Following [15] define an evolution operator V : S n,ν → S n,ν (which is called normalizedgonosomal operator) as V : x (cid:48) j = (cid:80) n,νi,k =1 γ ( f ) ik,j x i y k (cid:16)(cid:80) ni =1 x i (cid:17) (cid:16)(cid:80) νj =1 y j (cid:17) , j = 1 , . . . , ny (cid:48) l = (cid:80) n,νi,k =1 γ ( m ) ik,l x i y k (cid:16)(cid:80) ni =1 x i (cid:17) (cid:16)(cid:80) νj =1 y j (cid:17) , l = 1 , . . . , ν. (1.2) The main problem:
For given operator V and initial point z (0) ∈ S n,ν what ultimatelyhappens with the trajectory z ( m ) = V ( z ( m − ) , m = 1 , , . . . ? Does the limit lim m →∞ z ( m ) exist? If not what is the set of limit points of the sequence?In general, this is very difficult problem. In book [10] several recently obtained resultsrelated to this main problem are given.In this paper we consider the special case: n = ν = 2 and the following coefficients: γ ( f )11 , = a γ ( f )11 , = 0 γ ( m )11 , = b γ ( m )11 , = 0 γ ( f )12 , = 0 γ ( f )12 , = σ γ ( m )12 , = σ γ ( m )12 , = 0 γ ( f )21 , = 0 γ ( f )21 , = a γ ( m )21 , = b γ ( m )21 , = 0 γ ( f )22 , = 0 γ ( f )22 , = a γ ( m )22 , = 0 γ ( m )22 , = b. (1.3)Then corresponding evolution operator W : S , → S , is W : x (cid:48) = axu ( x + y )( u + v ) y (cid:48) = σ xv + ayu + ayv ( x + y )( u + v ) u (cid:48) = σ xv + bxu + byu ( x + y )( u + v ) v (cid:48) = byv ( x + y )( u + v ) , (1.4)where coefficients satisfy a + b = σ + σ = 1 , a, b, σ , σ > . REGULAR GONOSOMAL EVOLUTION OPERATOR 3
Remark 1.
From the probabilities (1.3) one can notice that type of females (resp. type of males) can be born only if both parents have type (resp. 2). Type of females (resp.type of males) can not be born if both parents have type (resp. ). For this operator W and arbitrarily initial point s (0) ∈ S , , we will study the trajectory { s ( m ) } ∞ m =0 , where s ( m ) = W m ( s (0) ) = W ( W ( ...W (cid:124) (cid:123)(cid:122) (cid:125) m ( s (0) )) ... ) . Fixed points
A point s is called a fixed point of the operator W if s = W ( s ). The set of all fixed pointsdenoted by Fix( W ).Let us find all the fixed points of W given by (1.4), i.e. we solve the following system ofequations for ( x, y, u, v ): x ( x + y )( u + v ) = axu,y ( x + y )( u + v ) = σ xv + ayu + ayv,u ( x + y )( u + v ) = σ xv + bxu + byu,v ( x + y )( u + v ) = byv. (2.1)If x = 0 then y (cid:54) = 0 and from the second equation of the system (2.1) we get y = a . Inaddition, the third and the fourth equations of the system (2.1) give u + v = b .If v = 0 then u (cid:54) = 0 and from the third equation of the system (2.1) we get u = b . Thesecond and the third equations of the system (2.1) give x + y = a .If xv (cid:54) = 0 then we come to ( x + y )( u + v ) = au,y ( x + y )( u + v ) = σ xv + ayu + ayv,u ( x + y )( u + v ) = σ xv + bxu + byu, ( x + y )( u + v ) = by. (2.2)The first and the second equations of the system (2.2) give σ x + ay = 0 . At the same time the third and the fourth equations of the system (2.2) give σ v + bu = 0 . Since a, b, σ , σ > x = y = u = v = 0,however this point is not in the space S , . Thus the set of all fixed points of operator (1.4)is Fix( W ) = F ∪ F , where F = (cid:110) (0 , a, u, v ) : u + v = b, u, v ∈ [0 , b ] (cid:111) and F = (cid:110) ( x, y, b,
0) : x + y = a, x, y ∈ [0 , a ] (cid:111) . A.T. ABSALAMOV AND U.A. ROZIKOV
Definition 1.
A fixed point s of the operator W is called hyperbolic if its Jacobian J at s has no eigenvalues on the unit circle. Definition 2.
A hyperbolic fixed point s is called:i) attracting if all the eigenvalues of the Jacobi matrix J ( s ) are less than 1 in absolutevalue;ii) repelling if all the eigenvalues of the Jacobi matrix J ( s ) are greater than 1 in absolutevalue;iii) a saddle otherwise. It is not hard to see that λ = 0, λ = 1, λ = 1 − vb and λ = 0, λ = 1, λ = 1 − xa areeigenvalues of the fixed points of the forms F and F respectively. By these definitionswe see that all fixed points of the operator (1.4) are nonhyperbolic fixed points.3. Limit set
Denote ∂S , = { t = ( x, y, u, v ) ∈ S , : xyuv = 0 } . Take any initial point t = ( x, y, u, v ) ∈ ∂S , . Consider the following subsets of ∂S , . E = (cid:8) ( x, y, u, v ) ∈ ∂S , : x = 0 (cid:9) , E = (cid:8) ( x, y, u, v ) ∈ ∂S , : y = 0 (cid:9) , E = (cid:8) ( x, y, u, v ) ∈ ∂S , : u = 0 (cid:9) , E = (cid:8) ( x, y, u, v ) ∈ ∂S , : v = 0 (cid:9) .If t = ( x, y, u, v ) ∈ E then W ( t ) ∈ F . If t = ( x, y, u, v ) ∈ E then W ( t ) ∈ F .When t = ( x, y, u, v ) ∈ E then W ( t ) ∈ E and W ( t ) ∈ F .When t = ( x, y, u, v ) ∈ E then W ( t ) ∈ E and W ( t ) ∈ F .Now we take any initial point t = ( x, y, u, v ) ∈ S , \ ∂S , .Introduce the following notations α = xx + y , β = vu + v , α (cid:48) = x (cid:48) x (cid:48) + y (cid:48) , β (cid:48) = v (cid:48) u (cid:48) + v (cid:48) , (3.1)which yields the nonlinear dynamical system V : α (cid:48) = α (1 − β )1 + ( p − αβ ,β (cid:48) = β (1 − α )1 + ( p − αβ (3.2)with the initial point ( α (0) , β (0) ) ∈ ∆, where∆ := { ( α, β ) ∈ R : 0 ≤ α ≤ , ≤ β ≤ } = [0 , , (3.3)and p = σ a , p = σ b . REGULAR GONOSOMAL EVOLUTION OPERATOR 5
There are three cases for p , p . 1 . p = p = 1 , . p > > p > , . p > > p > . (3.4)In order to find the fixed points of the operator (3.2) we solve the following system ofequations for ( α, β ) (cid:40) α (1 + ( p − αβ ) = α (1 − β ) ,β (1 + ( p − αβ ) = β (1 − α ) . (3.5)This system of equations gives us α · β = 0, that is s = ( α,
0) and s = (0 , β ) are fixedpoints for the operator (3.2) where α ≥ β ≥ α ( m +1) = α ( m ) (1 − β ( m ) )1 + ( p − α ( m ) β ( m ) ,β ( m +1) = β ( m ) (1 − α ( m ) )1 + ( p − α ( m ) β ( m ) . (3.6) Lemma 1.
For any initial point ( α, β ) ∈ [0 , it holds that ≤ α ( m +1) ≤ α ( m ) , ≤ β ( m +1) ≤ β ( m ) . In particular, the sequences α ( m ) = x ( m ) x ( m ) + y ( m ) , m ≥ and β ( m ) = v ( m ) u ( m ) + v ( m ) , m ≥ areconvergent.Proof. Since V : [0 , → [0 , and for any m ∈ N p − α ( m ) ∈ [min { , p } ; max { , p } ],1 + ( p − β ( m ) ∈ [min { , p } ; max { , p } ],1 + ( p − α ( m ) β ( m ) ∈ [min { , p } ; max { , p } ],1 + ( p − α ( m ) β ( m ) ∈ [min { , p } ; max { , p } ]then it holds that α ( m +1) − α ( m ) = − α ( m ) β ( m ) (1 + ( p − α ( m ) )1 + ( p − α ( m ) β ( m ) ≤ , and that β ( m +1) − β ( m ) = − α ( m ) β ( m ) (1 + ( p − β ( m ) )1 + ( p − α ( m ) β ( m ) ≤ . This completes the proof. (cid:3)
A.T. ABSALAMOV AND U.A. ROZIKOV
Theorem 1.
For any initial point ( x, y, u, v ) ∈ S , the sequence W m ( x, y, u, v ) = ( x ( m ) , y ( m ) , u ( m ) , v ( m ) ) is convergent and lim m →∞ x ( m ) · v ( m ) = 0 . Proof.
By Lemma 1 all trajectories of the operator (3.2) have a limit point and since theoperator is continuous, each trajectory converges to a fixed point s = ( α,
0) or s = (0 , β ).Therefore we have always α ( m ) · β ( m ) → , as m → ∞ . In a view of (1.4) and (3.1) we get x ( m +1) = ax ( m ) u ( m ) ( x ( m ) + y ( m ) )( u ( m ) + v ( m ) ) = aα ( m ) (1 − β ( m ) ) ,y ( m +1) = σ x ( m ) v ( m ) + ay ( m ) u ( m ) + ay ( m ) v ( m ) ( x ( m ) + y ( m ) )( u ( m ) + v ( m ) ) = σ α ( m ) β ( m ) + a (1 − α ( m ) ) ,u ( m +1) = σ x ( m ) v ( m ) + bx ( m ) u ( m ) + by ( m ) u ( m ) ( x ( m ) + y ( m ) )( u ( m ) + v ( m ) ) = σ α ( m ) β ( m ) + b (1 − β ( m ) ) ,v ( m +1) = by ( m ) v ( m ) ( x ( m ) + y ( m ) )( u ( m ) + v ( m ) ) = bβ ( m ) (1 − α ( m ) ) . (3.7)This completes the proof. (cid:3) Define the following sets: T = (cid:110) ( x, y, u, v ) ∈ S , : lim m →∞ x ( m ) = lim m →∞ v ( m ) = 0 (cid:111) ,T = (cid:110) ( x, y, u, v ) ∈ S , : lim m →∞ v ( m ) = 0 , lim m →∞ x ( m ) ∈ (0 , a ] (cid:111) ,T = (cid:110) ( x, y, u, v ) ∈ S , : lim m →∞ x ( m ) = 0 , lim m →∞ v ( m ) ∈ (0 , b ] (cid:111) . If t = ( x, y, u, v ) ∈ T , then lim m →∞ β ( m ) = lim m →∞ α ( m ) = 0 . (3.8)and (3.7) shows that for any initial point t = ( x, y, u, v ) ∈ T for the trajectories of theoperator (1.4) we have W m = ( x ( m ) , y ( m ) , u ( m ) , v ( m ) ) → (0 , a, b,
0) as m tends to ∞ . If t = ( x, y, u, v ) ∈ T , thenlim m →∞ β ( m ) = 0 and lim m →∞ α ( m ) = α ∈ (0 , . (3.9) REGULAR GONOSOMAL EVOLUTION OPERATOR 7
System of equations (3.7) shows that for any initial point t = ( x, y, u, v ) ∈ T for thetrajectories of the operator (1.4) we have W m = ( x ( m ) , y ( m ) , u ( m ) , v ( m ) ) → (cid:16) aα , a (1 − α ) , b, (cid:17) ∈ F as m tends to ∞ . If t = ( x, y, u, v ) ∈ T , thenlim m →∞ α ( m ) = 0 and lim m →∞ β ( m ) = β ∈ (0 , . (3.10)System of equations (3.7) shows that for any initial point t = ( x, y, u, v ) ∈ T for thetrajectories of the operator (1.4) we have W m = ( x ( m ) , y ( m ) , u ( m ) , v ( m ) ) → (cid:16) , a, b (1 − β ) , bβ (cid:17) ∈ F as m tends to ∞ . Therefore we have the following
Corollary 1.
For any initial point t = ( x, y, u, v ) ∈ S , the ω -limit set ω ( t ) of the operator (1.4) consists a single point and ω ( t ) ∈ { (0 , a, b, } if t = ( x, y, u, v ) ∈ T ,F if t = ( x, y, u, v ) ∈ T ,F if t = ( x, y, u, v ) ∈ T . (3.11) Definition 3.
An operator W is called regular if for any initial point s (0) ∈ S , , the limit lim m →∞ W m ( s (0) ) exists. The following is a corollary of Theorem 1.
Corollary 2.
The operator (1.4) is regular.
We would like to describe the sets T , T and T implicitly.3.1. Case 1.
Let we have p = p = 1 . Then operator (3.2) looks like: V : (cid:40) α (cid:48) = α − αββ (cid:48) = β − αβ (3.12)where ( α ; β ) ∈ ∆. s = ( α,
0) and s = (0 , β ) are non-hyperbolic fixed points of (3.12) with the eigenvalues λ = 1, λ = 1 − α ∈ [0 ,
1] and λ = 1, λ = 1 − β ∈ [0 ,
1] respectively.We say the set E is invariant respect to the operator V if V ( E ) ⊂ E . A.T. ABSALAMOV AND U.A. ROZIKOV
Lemma 2.
The following sets M = (cid:8) ( α, β ) ∈ [0 , : β = α (cid:9) M = (cid:8) ( α, β ) ∈ [0 , : β < α (cid:9) and M = (cid:8) ( α, β ) ∈ [0 , : β > α (cid:9) are invariant sets respect to the operator (3.12) .Proof. Straightforward. (cid:3)
We look for the invariant curves of the operator (3.12). Let β = g ( α ) be an invariantcurve then β (cid:48) = g ( α (cid:48) ) and to find invariant curve leads to solve the following iterativefunctional equation f ( α ) (cid:0) α − f ( α ) (cid:1) (1 − α ) = α (cid:0) f ( α ) − f ( f ( α )) (cid:1) (3.13)where f ( α ) = α (cid:0) − g ( α ) (cid:1) which is not identically zero.We solve (3.13) in the space C ∞ [0 , f (0) = 0. Moreover from f ∈ C ∞ [0 ,
1] we get f ( α ) = ∞ (cid:88) k =1 c k α k (3.14)and f ( f ( α )) = ∞ (cid:88) k =1 c k f k ( α ) = ∞ (cid:88) k =1 d k α k (3.15)where d k = k (cid:88) l =1 c l (cid:0) (cid:88) i + i + ... + i l = k c i · c i · ... · c i l (cid:1) . Theorem 2.
The solutions of the functional equation (3.13) are f ( α ) = α and f ( α ) = θα − α where θ is an arbitrary constant.In particular g ( α ) = 0 and g ( α ) = α + 1 − θ are the only invariant curves of the operator (3.12) .Proof. Substituting (3.14) and (3.15) to the (3.13) we obtain ∞ (cid:88) k =1 a k α k ≡ ∞ (cid:88) k =1 b k α k , (3.16)which is equivalently to a k = b k for all k = 1 , , ... (3.17) REGULAR GONOSOMAL EVOLUTION OPERATOR 9 where b k = k (cid:88) l =1 c l +1 (cid:0) (cid:88) i + i + ... + i l = k c i · c i · ... · c i l (cid:1) and a k = (cid:40) − − c + c if k = 1 ,c k +1 − c k if k = 2 , , ... (3.18)From identity of (3.16) for k = 1 it holds that(1 − c )(1 + c ) = 0 . For k = 2 we see that c (1 − c ) = c (1 + c ) . For k = 3 we see that c (1 − c ) = c (1 + c + 2 c c ) . These last three equations imply that c = 1 , c = 0 , c = 0 or c is arbitrary , c = − , c = 0 . Now we show by induction that c k = 0 for all k = 3 , , ... .Suppose c k = 0 for all k = 3 , , ...n . Then putting k = n , k = n + 1 and k = n + 2 in(3.17) we get c n +1 − c n = n (cid:88) l =1 c l +1 (cid:0) (cid:88) i + i + ... + i l = n c i · c i · ... · c i l (cid:1) c n +2 − c n +1 = n +1 (cid:88) l =1 c l +1 (cid:0) (cid:88) i + i + ... + i l = n +1 c i · c i · ... · c i l (cid:1) and c n +3 − c n +2 = n +2 (cid:88) l =1 c l +1 (cid:0) (cid:88) i + i + ... + i l = n +2 c i · c i · ... · c i l (cid:1) . These last three equations equivalent to c n +1 = c n +1 c n (3.19) c n +2 (1 − c n +11 ) = c n +1 (1 + c + nc c n − ) (3.20)and c n +3 (1 − c n +21 ) = c n +2 + c c n +2 + n !2! c n +1 c c n − + ( n + 1) c n +2 c c n (3.21)If c (cid:54) = ± c = − n is odd then (3.19) gives c n +1 = 0. If c = 1 then (3.20) gives c n +1 = 0, otherwise if c = − n is even then from (3.20) and (3.21) we come to c n +1 [(1 + (1 − n ) c )(1 + ( n + 2) c ) + n ! c ] = 0which shows again that c n +1 = 0. Thus for all k = 3 , , ... we have c k = 0. That is f ( α ) = α and f ( α ) = θα − α are solutions of the iterative functional equation (3.13), where θ is an arbitrary constant. This completes the proof. (cid:3)
So, we have proved that γ θ = (cid:8) ( α, β ) ∈ [0 , : β = g ( α ) = α + 1 − θ } , θ ∈ [0 , (cid:91) θ ∈ [0 , γ θ = M , (cid:91) θ ∈ (1 , γ θ = M , γ = M and γ θ ∩ γ θ = ∅ for any θ (cid:54) = θ . Thus it suffices to study the dynamical system on each invariant curve γ θ . We have thefollowing result (See Figure 1). Theorem 3.
The following assertions hold (i) If θ = 1 then for any initial point t = ( α, β ) ∈ M , (i.e. α = β ) we have lim m →∞ V ( m )1 ( α, β ) = lim m →∞ ( α ( m ) , β ( m ) ) = (0; 0) . (ii) If θ ∈ (1 , then for any initial point t = ( α, β ) ∈ γ θ we have lim m →∞ V ( m )1 ( α, β ) = lim m →∞ ( α ( m ) , β ( m ) ) = ( θ −
1; 0) . (iii) If θ ∈ [0 , then for any initial point t = ( α, β ) ∈ γ θ we have lim m →∞ V ( m )1 ( α, β ) = lim m →∞ ( α ( m ) , β ( m ) ) = (0; 1 − θ ) . Figure 1.
Dynamics of the operator (3.12) on the invariant lines γ θ . Thetrajectory converges to the fixed point on the intersection of the line and theaxes Oα or Oβ . REGULAR GONOSOMAL EVOLUTION OPERATOR 11
Going back to the old variables ( x, y, u, v ), when p = p = 1 we obtain σ = a , σ = b and Ω θ = (cid:8) ( x, y, u, v ) ∈ S , : vu + v = xx + y + 1 − θ (cid:9) is an invariant surface respect to the operator (1.4) and it holds that (cid:91) θ ∈ [0 , Ω θ = T = (cid:110) ( x, y, u, v ) ∈ S , : yv > xu (cid:111) , (cid:91) θ ∈ (1 , Ω θ = T = (cid:110) ( x, y, u, v ) ∈ S , : yv < xu (cid:111) , Ω = T = (cid:110) ( x, y, u, v ) ∈ S , : yv = xu (cid:111) and Ω θ ∩ Ω θ = ∅ for any θ (cid:54) = θ . Thus it suffices to study the dynamical system on each invariant surfaces Ω θ . As a corollaryof Theorem 3 we have the following Theorem 4.
The following assertions hold (i)
For any initial point t = ( x, y, u, v ) ∈ T , we have lim m →∞ W ( m ) ( x, y, u, v ) = lim m →∞ ( x ( m ) , y ( m ) , u ( m ) , v ( m ) ) = (0; a ; b ; 0) . (ii) If θ ∈ (1 , then for any initial point t = ( x, y, u, v ) ∈ Ω θ the following holds lim m →∞ W ( m ) ( x, y, u, v ) = lim m →∞ ( x ( m ) , y ( m ) , u ( m ) , v ( m ) ) = ( a ( θ − a (2 − θ ); b ; 0) . (iii) If θ ∈ [0 , then for any initial point t = ( x, y, u, v ) ∈ Ω θ the following holds lim m →∞ W ( m ) ( x, y, u, v ) = lim m →∞ ( x ( m ) , y ( m ) , u ( m ) , v ( m ) ) = (0; a ; bθ ; b (1 − θ )) . Corollary 3.
The operator (1.4) has infinitely many fixed points and for each such fixedpoint there is nonintersecting trajectories which converge to the fixed points.
Case 2.
Let we have p > > p > , or p > > p > . Lemma 3.
The set M = (cid:8) ( α, β ) ∈ ∆ : β ≥ α (cid:9) is an invariant set respect to the operator (3.2) when p > > p > .The set M = (cid:8) ( α, β ) ∈ ∆ : β ≤ α (cid:9) is an invariant set respect to the operator (3.2) when p > > p > .Proof. Straightforward. (cid:3)
In this case to find invariant curves for the operator (3.2) leads to solve the followingiterative functional equation f ( α ) (cid:0) α − f ( α ) (cid:1) (1 − α )[1 + ( p − f ( f ( α ))] = α (cid:0) f ( α ) − f ( f ( α )) (cid:1) [1 + ( p − α + ( p − p ) f ( α )] (3.22)where f ( α ) = α (cid:0) − g ( α ) (cid:1) \ [1 + ( p − αg ( α )] which is not identically zero.As above when we search the solution of the last functional equation in the space C ∞ [0 , ∞ (cid:88) k =1 a k α k ≡ ∞ (cid:88) k =1 b k α k , (3.23)which is equivalently to a k = b k for all k = 1 , , ... (3.24)where a k = k (cid:88) j =0 e j q k − j , b k = k (cid:88) j =0 n j m k − j and e j = − c if j = 0 , − (cid:80) jl =1 c l +1 (cid:0) (cid:80) i + i + ... + i l = j c i · c i · ... · c i l (cid:1) if j = 1 , , ..., k. (3.25) q j = j = 0 , ( p −
1) + ( p − p ) c if j = 1 , ( p − p ) c j if j = 2 , , ..., k. (3.26) n j = j = 0 , ( p − (cid:80) jl =1 c l (cid:0) (cid:80) i + i + ... + i l = j c i · c i · ... · c i l (cid:1) if j = 1 , , ..., k. (3.27) m j = − c if j = 0 , − c − c if j = 1 ,c j − c j +1 if j = 2 , , ..., k. (3.28)Substituting (3.25), (3.26), (3.27), (3.28) to the (3.24) then when k ≥ c k REGULAR GONOSOMAL EVOLUTION OPERATOR 13
Figure 2.
Dynamics of the operator (3.2) on invariant concave curves forthe case p > > p >
0. The trajectory converges to the fixed point on theintersection of the invariant curve and the axes Oα or Oβ . c k (1 − c k − ) = k − (cid:88) j =1 (cid:2) ( c k − j − − c k − j )( p − d j + ( p − p ) c k − j d (cid:48) j (cid:3) + c k − − ( p − d k − + (1 − c )( p − d k − (1 − c )( p − p ) c k + ( p − d (cid:48) k − − k − (cid:88) l =1 c l +1 (cid:0) (cid:88) i + i + ... + i l = k c i · c i · ... · c i l (cid:1) (3.29)where c k is the coefficient at (3.14) and d j = j (cid:88) l =1 c l (cid:0) (cid:88) i + i + ... + i l = j c i · c i · ... · c i l (cid:1) , d (cid:48) j = j (cid:88) l =1 c l +1 (cid:0) (cid:88) i + i + ... + i l = j c i · c i · ... · c i l (cid:1) . We were not able to solve these systems for the coefficients. Therefore the following isan open problem:
Open problem.
Describe all solutions of the functional equation (3.22).Numerical analysis shows that (see at the Figure 2 and Figure 3) in the cases p > > p > p > > p >
0) there are nonintersecting concave (resp.convex) invariant curves and the trajectory started on an invariant curve converges to theintersecting point of the invariant curve and the axes Oα or Oβ . These numerical analysis Figure 3.
Dynamics of the operator (3.2) on invariant convex curves forthe case p > > p >
0. The trajectory converges to the fixed point on theintersection of the invariant curve and the axes Oα or Oβ .and the above considered particular cases allowed us to make the following Conjecture. If p > > p > p > > p >
0) then for each fixed point p ∈ Fix( W ) there exists unique invariant surface Γ p ⊂ S , , such that for any initial point s (0) ∈ Γ p the limit of its trajectory (under operator (1.4)) converges to the fixed point p .Moreover, (cid:91) p ∈ Fix( W ) Γ p = S , . Conclusion
Let s (0) = ( x, y, u, v ) ∈ S , be an initial state, i.e. the probability distribution on the setof female and male types.The following are interpretations of our results: • The set of all fixed points is subset of the boundary of S , means that at least onetype of female or male in future of population will surely disappear. • The existence of invariant curves (in particular lines) means that if states of thepopulation initially satisfied a relation (described the invariant set) then the futureof the population remains in the same relation. • Regularity of the operator means that for any initial state of the population we canexplicitly determine its limit (final) state.
REGULAR GONOSOMAL EVOLUTION OPERATOR 15 • For any s (0) ∈ T as time goes to infinity the type 1 of female and type 2 of maleswill disappear (die). • For any s (0) ∈ T as time goes to infinity the type 2 of males will disappear. • For any s (0) ∈ T as time goes to infinity the type 1 of females will disappear. References [1] Absalamov A.T., Rozikov U.A.
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UkraineMath. Jour. (7) (2011), 985–998.[15] Rozikov U.A., Varro R. Dynamical systems generated by a gonosomal evolution operator.
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A.T. Absalamov, Samarkand State University, Boulevard str., 140104, Samarkand, Uzbek-istan.
Email address : [email protected] U.A. Rozikov a,b,ca
V.I.Romanovskiy Institute of Mathematics of Uzbek Academy of Sciences; b AKFA University, 1st Deadlock 10, Kukcha Darvoza, 100095, Tashkent, Uzbekistan; c Faculty of Mathematics, National University of Uzbekistan.
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