A discrete-time dynamical system of wild mosquito population with Allee effects
AA DISCRETE-TIME DYNAMICAL SYSTEM OF WILD MOSQUITOPOPULATION WITH ALLEE EFFECTS
U.A. ROZIKOV, Z.S. BOXONOV
Abstract.
We study a discrete-time dynamical system of wild mosquito population withparameters: β - the birth rate of adults; α - maximum emergence rete; µ > γ - Allee effects. We prove that if γ ≥ α ( β − µ ) µ then the mosquito populationdies and if γ < α ( β − µ ) µ holds then extinction or survival of the mosquito population dependson their initial state. Introduction
Mosquito population control is a vital public-health practice throughout the world andespecially in the tropics because mosquitoes spread many diseases, such as malaria and variousviruses.Today several kind of mathematical models of mosquito population are known (see [1], [2],[4], [8], [13], [10] and references therein).A mathematical model of mosquito dispersal in continuous time was investigated in [9].Recently, in [13] a discrete-time dynamical system, generated by an evolution operator of thismosquito population is studied.It is known that during a lifetime mosquitoes undergo complete metamorphosis goingthrough four distinct stages of development: egg, larva, pupa and adult [1], [2], [4], [6].In [7] continuous-time model of mosquito population with Allee effects is studied, here weconsider a discrete-time dynamical system of this model. Consider a wild mosquito populationwithout the presence of sterile mosquitoes and in a simplified stage-structured population,one groups the three aquatic stages into the larvae class by x , and divide the mosquitopopulation into the larvae class and the adults, denoted by y . Moreover, assume that thedensity dependence exists only in the larvae stage [7].Denote the birth rate, i.e., the oviposition rate of adults by β ( · ); the rate of emergencefrom larvae to adults by a function of the larvae with the form of α (1 − k ( x )), where α > ≤ k ( x ) ≤
1, with k (0) = 0 , k (cid:48) ( x ) >
0, and lim x →∞ k ( x ) = 1,is the functional response due to the intraspecific competition [8]. Moreover, we assume thedeath rate of larvae be a linear function, denoted by d + d x , and the death rate of adultsbe constant, denoted by µ . Then, in the absence of sterile mosquitoes, we get the following Mathematics Subject Classification.
Key words and phrases. mosquito; population; Allee effects; fixed point; limit point. a r X i v : . [ m a t h . D S ] F e b U.A. ROZIKOV, Z.S. BOXONOV system of equations: dxdt = β ( · ) y − α (1 − k ( x )) x − ( d + d x ) x, dydt = α (1 − k ( x )) x − µy. (1.1)We further assume a functional response for k ( x ), as in [8], in the form k ( x ) = x x . In [7],[8] the dynamical system (1.1) was studied for β ( · ) = β (i.e. when mosquito adultshave no difficulty to find their mates such that no Allee effects are concerned) and thediscrete-time version of this model was considered in [11] and [12].A component Allee effect is defined as a decrease in any component of fitness with de-creasing population size or density. A decrease in the probability of a female mating withdecreasing male density is therefore a component Allee effect, and is generally referred to asa ”mate-finding Allee effect” [3].In the case where adult mosquitoes have difficulty in finding their mates, Allee effects areincluded and the adult birth rate is given by β ( · ) = βyγ + y , where γ is the Allee effect constant.Then stage-structured wild mosquito population model is given by [7]: dxdt = βy γ + y − αx x − ( d + d x ) x, dydt = αx x − µy. (1.2)In this paper (as in [13], [14]) we study the discrete time dynamical systems associated tothe system (1.2). Define the operator W : R → R by x (cid:48) = βy γ + y − αx x − ( d + d x ) x + x,y (cid:48) = αx x − µy + y, (1.3)where α > , β > , γ > , µ > , d ≥ , d ≥ . In this paper we consider the operator W (defined by (1.3)) for the case d = d = 0 andour aim is to study trajectories z ( n ) = W ( z ( n − ), n ≥ z (0) = ( x (0) , y (0) ).Note that this system, (1.3), for the case when d (cid:54) = 0 or d (cid:54) = 0 is not studied yet.2. Dynamical system generated by the operator (1.3)
We assume d = d = 0 (2.1)then (1.3) has the following form W : x (cid:48) = βy γ + y − αx x + x,y (cid:48) = αx x − µy + y. (2.2) DISCRETE-TIME DYNAMICAL SYSTEM OF MOSQUITO POPULATION 3
It is easy to see that if 0 < α ≤ , β > , γ > , < µ ≤ R = { ( x, y ) ∈ R : x ≥ , y ≥ } to itself.2.1. Fixed points.
A point z ∈ R is called a fixed point of W if W ( z ) = z .For fixed point of W the following holds. Proposition 1.
The fixed points for (2.2) are as follows: • If β ≤ µ (1 + γµα ) then the operator (1.3) has a unique fixed point z = (0 , . • If β > µ (1 + γµα ) then mapping (1.3) has two fixed points with z = (0 , , z = ( γµ α ( β − µ ) − γµ , γµβ − µ ) . Proof.
We need to solve x = βy γ + y − αx x + x,y = αx x − µy + y (2.4)It is easy to see that x = 0 , y = 0 and x = γµ α ( β − µ ) − γµ , y = γµβ − µ are solution to (2.4). If β ≤ µ (1 + γµα ) then x / ∈ R , otherwise x ∈ R . (cid:3) The type of the fixed point.
Now we shall examine the type of the fixed point.
Definition 1. (see [5] ) A fixed point s of an operator W is called hyperbolic if its Jacobian J at s has no eigenvalues on the unit circle. Definition 2. (see [5] ) A hyperbolic fixed point s is called: attracting if all the eigenvalues of the Jacobi matrix J ( s ) are less than 1 in absolutevalue; repelling if all the eigenvalues of the Jacobi matrix J ( s ) are greater than 1 inabsolute value; a saddle otherwise. To find the type of a fixed point of the operator (2.2) we write the Jacobi matrix: J ( z ) = J W = (cid:32) − α (1+ x ) βy (2 γ + y )( γ + y ) α (1+ x ) − µ (cid:33) . The eigenvalues of the Jacobi matrix at the fixed point (0 .
0) are as follows λ = 1 − α, λ = 1 − µ By (2.3) we have 0 ≤ λ , < x ∗ = γµ α ( β − µ ) − γµ , y ∗ = γµβ − µ . (2.5) U.A. ROZIKOV, Z.S. BOXONOV
We calculate eigenvalues of Jacobian matrix at the fixed point ( x ∗ , y ∗ ). If we denote 1 − λ = Λthen we obtain Λ − ( µ + A )Λ + A ( µ − B ) = 0 , where A = α (1+ x ∗ ) , B = βy ∗ (2 γ + y ∗ )( γ + y ∗ ) . − ¯ λ , = Λ , = 12 ( µ + A ± (cid:112) ( µ − A ) + 4 AB ) . (2.6)The inequality | ¯ λ , | < < Λ , < . Since µ + A < < Λ , < (cid:40) µ + A ± (cid:112) ( µ − A ) + 4 AB > µ + A ± (cid:112) ( µ − A ) + 4 AB < . From the first inequality µ + A − (cid:112) ( µ − A ) + 4 AB > µ > B .If we consider B = βy ∗ (2 γ + y ∗ )( γ + y ∗ ) = β (1 − γ ( γ + y ∗ ) ) = β − β ( β − µ ) , β > µ (1 + γµα )then µ > β − β ( β − µ ) , or µ <
0. It is a contradiction for (2.3). Therefore, the fixed point( x ∗ , y ∗ ) is not attracting.The inequality | ¯ λ , | > , < , > . For values of the parameters given in (2.3) the inequalities µ + A + (cid:112) ( µ − A ) + 4 AB < ,µ + A − (cid:112) ( µ − A ) + 4 AB > x ∗ , y ∗ ) is not repelling.Thus the type of fixed points the following proposition holds. Proposition 2.
The type of the fixed points for (2.2) are as follows: i) (0 , is attracting; ii) if β > µ (1 + γµα ) then ( x ∗ , y ∗ ) is saddle. The limits of trajectories.
Let β ≤ µ (1 + γµα ) (2.7)The following theorem describes the trajectory of any point ( x (0) , y (0) ) in R . Theorem 1.
For the operator W given by (2.2) (i.e. under condition (2.7)) and for anyinitial point ( x (0) , y (0) ) ∈ R the following hold lim n →∞ x ( n ) = 0 , lim n →∞ y ( n ) = 0 , where ( x ( n ) , y ( n ) ) = W n ( x (0) , y (0) ) , with W n is n -th iteration of W . DISCRETE-TIME DYNAMICAL SYSTEM OF MOSQUITO POPULATION 5
Proof. If β ≤ µ (1 + γµα ) then there exists k ≥ β · k = µ (1 + γµα ). Denote c ( n ) = x ( n ) + y ( n ) and c ( n )0 = k · x ( n ) + y ( n ) , where x ( n ) , y ( n ) defined by the following x ( n ) = β ( y ( n − ) γ + y ( n − − αx ( n − x ( n − + x ( n − ,y ( n ) = αx ( n − x ( n − − µy ( n − + y ( n − . (2.8) Lemma 1.
There exists n such that the sequence y ( n ) is less than αµ for n ≥ n .Proof. Assume that all values of y ( n ) are not less than αµ . Then y ( n +1) − y ( n ) = αx ( n ) x ( n ) − µy ( n ) < α − µy ( n ) = µ ( αµ − y ( n ) ) ≤ . So y ( n ) is a decreasing sequence. Since y ( n ) is decreasing and bounded from below we have:lim n →∞ y ( n ) ≥ αµ . (2.9)We estimate y ( n ) by the following: y ( n ) = αx ( n − x ( n − + (1 − µ ) y ( n − < α + (1 − µ ) y ( n − < α + (1 − µ )( α + (1 − µ ) y ( n − ) < α + α (1 − µ ) + (1 − µ ) ( α + (1 − µ ) y ( n − ) < ... < α + α (1 − µ ) + α (1 − µ ) + ... + α (1 − µ ) n − +(1 − µ ) n y (0) = αµ + (1 − µ ) n ( y (0) − αµ ) . Thus y ( n ) < αµ + (1 − µ ) n ( y (0) − αµ ). Consequentlylim n →∞ y ( n ) < αµ . (2.10)By (2.9) and (2.10) our assumption is false. Hence, there exists n such that y ( n ) is less than αµ . If y ( n − < αµ then y ( n ) < αµ . Indeed, y ( n ) = (1 − µ ) y ( n − + αx ( n − x ( n − < (1 − µ ) αµ + αx ( n − x ( n − = αµ − α (1 − x ( n − x ( n − ) < αµ . (cid:3) U.A. ROZIKOV, Z.S. BOXONOV
By Lemma 1 we have y ( n − < αµ and β ≤ µ (1 + γµα ). Then( β − µ ) y ( n − − γµ < , (1 + γµα ) y ( n − γ + y ( n − − < . (2.11)By using (2.11) to the equalities c ( n ) and c ( n )0 we obtain the followings. c ( n ) = x ( n ) + y ( n ) = x ( n − + y ( n − + y ( n − γ + y ( n − (( β − µ ) y ( n − − γµ ) < x ( n − + y ( n − = c ( n − ,c ( n )0 = k · x ( n ) + y ( n ) = k · x ( n − + y ( n − + (1 − k ) αx ( n − x ( n − + µy ( n − ((1 + γµα ) y ( n − γ + y ( n − − < k · x ( n − + y ( n − = c ( n − . Hence both sequences { c ( n ) } and { c ( n )0 } are monotone and bounded, i.e.,0 < ... < c ( n ) < c ( n − < ... < c (0) , < ... < c ( n )0 < c ( n − < ... < c (0)0 . Thus { c ( n ) } and { c ( n )0 } have limit points, denote the limits by c ∗ and c ∗ respectively. Conse-quently, the following limits exist˜ x = lim n →∞ x ( n ) = 11 − k lim n →∞ ( c ( n ) − c ( n )0 ) = 11 − k ( c ∗ − c ∗ ) , ˜ y = lim n →∞ y ( n ) = c ∗ − ˜ x. and by (2.8) we have ˜ x = β ˜ y γ + ˜ y − α ˜ x x + ˜ x, ˜ y = α ˜ x x − µ ˜ y + ˜ y, i.e., ˜ x = 0 , ˜ y = 0 . (cid:3) Dynamics on invariant sets.
A set A is called invariant with respect to W if W ( A ) ⊂ A .Denote Ω = { ( x, y ) ∈ R , ≤ x ≤ x ∗ , ≤ y ≤ y ∗ } \ { ( x ∗ , y ∗ ) } Ω = { ( x, y ) ∈ R , x ∗ ≤ x, y ∗ ≤ y } \ { ( x ∗ , y ∗ ) } where ( x ∗ , y ∗ ) is the fixed point defined by (2 . W given by (2.2) in the sets Ω , Ω undercondition β > µ (1 + γµα ) . Lemma 2.
The sets Ω and Ω are invariant with respect to W . DISCRETE-TIME DYNAMICAL SYSTEM OF MOSQUITO POPULATION 7
Proof. Let 0 ≤ x ≤ x ∗ , ≤ y ≤ y ∗ . Then x (cid:48) − x ∗ = βy γ + y + x (1 − α x ) − x ∗ ≤ β ( y ∗ ) γ + y ∗ + x ∗ (1 − α x ) − x ∗ = β ( y ∗ ) γ + γµβ − µ − αx ∗ x ≤ β − µγ ( y ∗ ) − αx ∗ x ∗ = β − µγ ( y ∗ ) − µy ∗ = y ∗ ( β − µγ · γµβ − µ − µ ) = 0 .y ∗ − y (cid:48) = y ∗ − (1 − µ ) y − αx x ≥ y ∗ − (1 − µ ) y ∗ − αx ∗ x ∗ = µy ∗ − αx ∗ x ∗ = 0 . Thus ( x (cid:48) , y (cid:48) ) ∈ W (Ω ) ⊂ Ω Let x ≥ x ∗ , y ≥ y ∗ . Then x ∗ − x (cid:48)(cid:48) = x ∗ − βy γ + y − x (1 − α x ) ≤ x ∗ − β ( y ∗ ) γ + y ∗ − x ∗ (1 − α x )= αx ∗ x − β − µγ y ∗ ≤ αx ∗ x ∗ − β − µγ y ∗ = y ∗ ( µ − β − µγ γµβ − µ ) = 0 .y (cid:48)(cid:48) − y ∗ = αx x + (1 − µ ) y − y ∗ ≥ αx ∗ x ∗ + (1 − µ ) y ∗ − y ∗ = 0 . Thus ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) ∈ W (Ω ) ⊂ Ω . (cid:3) The following theorem describes the trajectory of any point ( x (0) , y (0) ) in invariant sets. Theorem 2.
For the operator W given by (2.2) (i.e. under condition (2.3)), if β > µ (1+ γµα ) then for any initial point ( x (0) , y (0) ) , the following hold lim n →∞ x ( n ) = (cid:40) , if ( x (0) , y (0) ) ∈ Ω , + ∞ , if ( x (0) , y (0) ) ∈ Ω lim n →∞ y ( n ) = (cid:40) , if ( x (0) , y (0) ) ∈ Ω , αµ , if ( x (0) , y (0) ) ∈ Ω where ( x ( n ) , y ( n ) ) = W n ( x (0) , y (0) ) .Proof. Adding x ( n ) and y ( n ) we get (see (2.5)) x ( n ) + y ( n ) = x ( n − + y ( n − − ( β − µ ) y ( n ) γ + y ( n ) ( y ∗ − y ( n ) ) (2.12) Lemma 3.
The sequence y ( n ) is bounded.Proof. Is easily deduced from Lemma 1. (cid:3)
Lemma 4.
For sequences x ( n ) and y ( n ) in the set Ω the following statements hold: For any n ∈ N the inequalities x ( n ) < x ( n +1) and y ( n ) < y ( n +1) can not be satisfied atthe same time. U.A. ROZIKOV, Z.S. BOXONOV If x ( m − > x ( m ) , y ( m − > y ( m ) for some m ∈ N then x ( m ) > x ( m +1) , y ( m ) > y ( m +1) . If x ( m − > x ( m ) , y ( m − < y ( m ) for any m ∈ N then for β > µ (1+ γµα ) the inequalities x ( m ) > x ( m +1) , y ( m ) < y ( m +1) can not be satisfied at the same time. If x ( m − < x ( m ) , y ( m − > y ( m ) for any m ∈ N then for β > µ (1+ γµα ) the inequalities x ( m ) < x ( m +1) , y ( m ) > y ( m +1) can not be satisfied at the same time.Proof. Let ( x (0) , y (0) ) ∈ Ω . For each n ∈ N , x ( n ) ≤ x ∗ and y ( n ) ≤ y ∗ .
1) From (2.12) we get ( x ( n ) − x ( n − ) + ( y ( n ) − y ( n − ) < . Consequently, x ( n ) < x ( n +1) and y ( n ) < y ( n +1) can not be satisfied at the same time.2) By x ( m − − x ( m ) > y ( m − − y ( m ) > x ( m − (1 − α x ( m − ) − x ( m ) (1 − α x ( m ) ) > , x ( m − x ( m − > x ( m ) x ( m ) and y ( m − γ + y ( m − > y ( m ) γ + y ( m ) . Then x ( m ) − x ( m +1) = β ( y ( m − γ + y ( m − − y ( m ) γ + y ( m ) ) + x ( m − (1 − α x ( m − ) − x ( m ) (1 − α x ( m ) ) > ,y ( m ) − y ( m +1) = α ( x ( m − x ( m − − x ( m ) x ( m ) ) + (1 − µ )( y ( m − − y ( m ) ) > .
3) Assume in case when x ( m − > x ( m ) , y ( m − < y ( m ) hold for any m ∈ N then for β > µ (1 + γµα ) the inequalities x ( m ) > x ( m +1) , y ( m ) < y ( m +1) are satisfied at the sametime. Then since x ( n ) is decreasing and bounded; y ( n ) is increasing and bounded (seeLemma 3) there exist their limits ˜ x , ˜ y (cid:54) = 0 respectively. By (2.8) we obtain (cid:40) β ˜ y γ +˜ y = α ˜ x xα ˜ x x = µ ˜ y i.e. ˜ x = 0 , ˜ y = 0 . This contradiction shows that if for any m ∈ N one has x ( m − >x ( m ) , y ( m − < y ( m ) then for β > µ (1 + γµα ) the inequalities x ( m ) > x ( m +1) , y ( m )
Lemma 5.
For sequences x ( n ) and y ( n ) in the set Ω the following statements hold: For any n ∈ N the inequalities x ( n ) > x ( n +1) and y ( n ) > y ( n +1) can not be satisfied atthe same time. If x ( m − < x ( m ) , y ( m − < y ( m ) for some m ∈ N then x ( m ) < x ( m +1) , y ( m ) < y ( m +1) . DISCRETE-TIME DYNAMICAL SYSTEM OF MOSQUITO POPULATION 9 If x ( m − > x ( m ) , y ( m − < y ( m ) for any m ∈ N then for β > µ (1+ γµα ) the inequalities x ( m ) > x ( m +1) , y ( m ) < y ( m +1) can not be satisfied at the same time. If x ( m − < x ( m ) , y ( m − > y ( m ) for any m ∈ N then for β > µ (1+ γµα ) the inequalities x ( m ) < x ( m +1) , y ( m ) > y ( m +1) can not be satisfied at the same time.Proof. Let ( x (0) , y (0) ) ∈ Ω . For each n ∈ N , x ( n ) ≥ x ∗ and y ( n ) ≥ y ∗ .
1) From (2.12) we get ( x ( n ) − x ( n − ) + ( y ( n ) − y ( n − ) > . Consequently, x ( n ) > x ( n +1) and y ( n ) > y ( n +1) can not be satisfied at the same time.2) By x ( m ) − x ( m − > y ( m ) − y ( m − > x ( m ) (1 − α x ( m ) ) − x ( m − (1 − α x ( m − ) > , x ( m ) x ( m ) > x ( m − x ( m − and y ( m ) γ + y ( m ) > y ( m − γ + y ( m − . Then x ( m +1) − x ( m ) = βy ( m ) γ + y ( m ) − βy ( m − γ + y ( m − + x ( m ) (1 − α x ( m ) ) − x ( m − (1 − α x ( m − ) > ,y ( m +1) − y ( m ) = α ( x ( m ) x ( m ) − x ( m − x ( m − ) + (1 − µ )( y ( m ) − y ( m − ) > .
3) Similarly to the proof of part 3 of Lemma 4.4) Assume if for any m ∈ N the inequalities x ( m − < x ( m ) , y ( m − > y ( m ) hold then for β > µ (1 + γµα ) the inequalities x ( m ) < x ( m +1) , y ( m ) > y ( m +1) are satisfied at the sametime, i.e. x ( m ) is increasing and y ( m ) is decreasing. Let∆ ( m ) = ( x ( m +1) − x ( m ) ) + ( y ( m +1) − y ( m ) ) = ( β − µ ) y ( m ) γ + y ( m ) ( y ( m ) − y ∗ ) . Since { y ( m ) } is decreasing, ∆ ( m ) > β > µ (1 + γµα ) we conclude that thesequence { ∆ ( m ) } is decreasing and bounded from below. Thus { ∆ ( m ) } has a limitand since y ( m ) has limit we conclude that x ( m ) has a finite limit. By (2.8) we havelim m →∞ x ( m ) = 0 , lim m →∞ y ( m ) = 0 . But this is a contradiction to lim m →∞ x ( m ) (cid:54) = 0. This completes proof of part 4. (cid:3) Lemma 6. If x ( n ) and y ( n ) in Ω , then there exists n ∈ N such that x ( n ) and y ( n ) aredecreasing for n ≥ n , if x ( n ) and y ( n ) in Ω , then there exists m ∈ N such that x ( n ) and y ( n ) are increasing for n ≥ m .Proof. Monotonicity of x ( n ) and y ( n ) follow from Lemma 4 and Lemma 5. (cid:3) Lemma 7. If x ( n ) in the set Ω then x ( n ) is unbounded from above. Proof.
There exists n such that the sequences x ( n ) is increasing for n ≥ n . Consider ( x ( n +1) − x ( n ) ) + ( y ( n +1) − y ( n ) ) = ( β − µ ) y ( n γ + y ( n ( y ( n ) − y ∗ )( x ( n +2) − x ( n +1) ) + ( y ( n +2) − y ( n +1) ) = ( β − µ ) y ( n γ + y ( n ( y ( n +1) − y ∗ ) ... ( x ( n − − x ( n − ) + ( y ( n − − y ( n − ) = ( β − µ ) y ( n − γ + y ( n − ( y ( n − − y ∗ )( x ( n ) − x ( n − ) + ( y ( n ) − y ( n − ) = ( β − µ ) y ( n − γ + y ( n − ( y ( n − − y ∗ ) (2.13)Adding equations of (2.13) we get( x ( n ) − x ( n ) ) + ( y ( n ) − y ( n ) )= ( β − µ ) (cid:32) y ( n ) γ + y ( n ) ( y ( n ) − y ∗ ) + ... + y ( n − γ + y ( n − ( y ( n − − y ∗ ) (cid:33) (2.14)Let y ( n ) (see Lemma 3) is bounded by θ . By (2.14) we have x ( n ) > x ( n ) + y ( n ) − θ + ( β − µ )( n − n )( y ( n ) − y ∗ ) · y ( n ) γ + y ( n ) . For β > µ (1 + γµα ) fromlim n →∞ (cid:32) x ( n ) + y ( n ) − θ + ( β − µ )( n − n )( y ( n ) − y ∗ ) · y ( n ) γ + y ( n ) (cid:33) = + ∞ it follows that x ( n ) is not bounded from above. (cid:3) If ( x (0) , y (0) ) ∈ Ω then by Lemma 6 there exist their limits ˜ x , ˜ y respectively. By (2.8) wehave lim n →∞ x ( n ) = 0 , lim n →∞ y ( n ) = 0 . If ( x (0) , y (0) ) ∈ Ω then for β > µ (1 + γµα ) the sequence y ( n ) has limit ˜ y (see Lemma 3).Consequently, by (2.8) and lim n →∞ x ( n ) = + ∞ we get ˜ y = αµ . Theorem is proved. (cid:3) On the set R \ (Ω (cid:83) Ω ) . In the following examples, we show trajectories of initial points from the set R \ (Ω (cid:83) Ω ).Under conditions β > µ (1 + γµα ), the parameters are α = 0 . , β = 0 . , γ = 2 , µ = 0 .
4. Thenby (2.5), we get x ∗ = 4, y ∗ = 1 . Example 1.
If the initial point is x (0) = 0 . , y (0) = 4 then the trajectory of system (2.2) isshown in the Fig.1, i.e., lim n →∞ x ( n ) = 0 , lim n →∞ y ( n ) = 0 . If the initial point is x (0) = 0 . , y (0) = 5 then the trajectory of system (2.2) is shown inthe Fig.1. In this case the first coordinate of the trajectory goes to infinite and the secondcoordinate has limit point approximately , i.e., lim n →∞ x ( n ) = + ∞ , lim n →∞ y ( n ) = αµ = 2 . DISCRETE-TIME DYNAMICAL SYSTEM OF MOSQUITO POPULATION 11
Example 2.
If the initial point is x (0) = 5 . , y (0) = 0 . then the trajectory of system (2.2)is shown in the Fig.2, i.e., it converges to (0 , .If the initial point is x (0) = 7 , y (0) = 0 . then the trajectory of system (2.2) is shown inthe Fig.2. In this case the first coordinate of the trajectory goes to infinite and the secondcoordinate has limit point approximately . Figure 1. x (0) = 0 . , y (0) = 4 and x (0) = 0 . , y (0) = 53. Biological interpretations
Each point (vector) z = ( x ; y ) ∈ R can be considered as a state (a measure) of themosquito population.Let us give some interpretations of our main results:(a) (Case Theorem 1) Since (2.7) we have γ ≥ α ( β − µ ) µ . Under this condition on γ (i.e. onAllee effects), the mosquito population dies;(b) (Case Theorem 2) If the inequality γ < α ( β − µ ) µ holds for Allee effects γ , then extinctionor survival of the mosquito population depends on their initial state. References [1] L. Alphey, M. Benedict, R. Bellini, G.G. Clark, D.A. Dame, M.W. Service, and S.L. Dobson,
Sterile-insect methods for control of mosquito-borne diseases : An analysis, Vector Borne Zoonotic Dis. (2010),295–311.[2] A.C. Bartlettand, R. T.Staten, Sterile Insect Release Method and Other Genetic Control Strategies , Rad-cliffe’s IPM World Textbook, 1996.[3] X. Fauvergue
A review of mate-finding Allee effects in insects: from individual behavior to populationmanagement , https://doi.org/10.1111/eea.12021
Figure 2. x (0) = 5 . , y (0) = 0 . x (0) = 7 , y (0) = 0 . [4] N. Becker, Mosquitoes and Their Control , Kluwer Academic/Plenum, New York, 2003.[5] R.L. Devaney,
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World Sci. Publ. Singapore.2020, 460 pp.[11] Z.S. Boxonov, U.A. Rozikov,
A discrete-time dynamical system of stage-structured wild and sterilemosquito population . arXiv:2002.11995.[12] Z.S. Boxonov, U.A. Rozikov,
Dynamical system of a mosquito population with distinct birth-death rates .arXiv:2007.03270. To appear in Jour. Appl. Nonlinear Dynamics.[13] U.A. Rozikov, M.V. Velasco,
A discrete-time dynamical system and an evolution algebra of mosquitopopulation . Jour. Math. Biology. (4) (2019), 1225–1244.[14] U.A. Rozikov, S.K. Shoyimardonov, On ocean ecosystem discrete time dynamics generated by (cid:96) -Volterraoperators . Inter. Jour. Biomath. (2) (2019), 1950015, (24 pages). U. A. RozikovV.I.Romanovskiy Institute of Mathematics of Uzbek Academy of Sciences;AKFA University, 1st Deadlock 10, Kukcha Darvoza, 100095, Tashkent, Uzbekistan;Faculty of Mathematics, National University of Uzbekistan.
Email address : [email protected] Z. S. Boxonov, V.I.Romanovskiy Institute of Mathematics of Uzbek Academy of Sciences,Tashkent, Uzbekistan.
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