Dynamics of a modified Nicholson-Bailey host-parasitoid model: [Doc 357]

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

In this paper, we study the qualitative behavior of the following modified Nicholson-Bailey host-parasitoid model: ...... where a, b, c, d and the initial conditions ......, ...... are positive real numbers. More precisely, we investigate the boundedness character, existence and uniqueness of a positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point, and the rate of convergence of positive solutions of the system. Some numerical examples are also given to verify our theoretical results.

Khan and Qureshi Advances in Dierence Equations (2015) 2015:23 DOI 10.1186/s13662-015-0357-2

*Correspondence: mailto:[email protected]

Web End [email protected] Department of Mathematics, University of Azad Jammu and Kashmir, Muzaarabad, 13100, Pakistan

R E S E A R C H Open Access

Dynamics of a modied Nicholson-Bailey host-parasitoid model

Abdul Qadeer Khan* and Muhammad Naeem Qureshi

Abstract

In this paper, we study the qualitative behavior of the following modied Nicholson-Bailey host-parasitoid model:

xn+1 = bxneayn

1 + dxn , yn+1 = cxn(1 eayn),

where a, b, c, d and the initial conditions x0, y0 are positive real numbers. More precisely, we investigate the boundedness character, existence and uniqueness of a positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point, and the rate of convergence of positive solutions of the system. Some numerical examples are also given to verify our theoretical results. MSC: 39A10; 40A05

Keywords: modied Nicholson-Bailey model; boundedness; local stability; global character; rate of convergence

1 Introduction

Many ecological models are governed by dierential as well as dierence equations. In particular, ecological models with non-overlapping populations are better formulated as discrete dynamical systems compared to the continuous time models. These models have been extensively studied in recent years because of their wide applicability to the study of population dynamics [, ]. In fact, in the case of discrete dynamical systems, one has more ecient computational results for numerical simulations and also has rich dynamics as compared to the continuous ones. In recent years, several papers have been published on the mathematical models of biology that discuss the system of dierence equations generated from the associated system of dierential equations as well as the associated numerical methods [, ]. In mathematical biology, the model such as the host-parasitoid has attracted many researchers during the last few decades. Usually, the biologists believe that a unique, positive, locally asymptotically stable equilibrium point in an ecological system is very important []. Therefore, it is pertinent to nd conditions which may guarantee the global stability of a positive equilibrium point, if it exist, for the given system. See [] for introduction to mathematical models in biological sciences.

The prime example of an ecologically interesting discrete-time model for interacting populations is the Nicholson-Bailey model for host-parasitoid dynamics. Parasitoids are insect species whose larvae develop by feeding on the bodies of other arthropods, usually

2015 Khan and Qureshi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Khan and Qureshi Advances in Dierence Equations (2015) 2015:23 Page 2 of 15

killing them. Larvae emerge from the host and develop into free-living adults. The adults then lay their eggs in a subsequent generation of hosts. Most parasitoid larvae require a specic life-stage of the host, so parasitoid and host generations are linked to one another. Consequently host-parasitoid models often use a discrete time step corresponding to the common generation length of host and parasitoid. The classic model was derived by Nicholson and Bailey (). The assumptions of the Nicholson-Bailey model are as follows:(i) Hosts are distributed at random, at density xn per unit area in generation n.(ii) Parasitoids search at random and independently, each having an area of discovery a, and lay an egg in each host found.

(iii) Each parasitized host gives rise to one new parasitoid in generation n + .(iv) Each unparasitized host gives rise to b > new hosts in generation n + .

Each parasitoid attacks the hosts found in a units of area, so the expected number of hosts attacked by each parasitoid is ax. The expected total number of attacks is axy. The total number of attacks can also be written as the sum over hosts of the number of attacks on each host. All hosts have the same expected number of attacks, so the expected number of attacks on any given host must be ay. Under the assumptions listed above, the number of eggs per host has a Poisson distribution. Consequently, the expected fraction of hosts that are not parasitized is the probability that a Poisson random variable with mean ay takes the value zero. The resulting population dynamics are

xn+ = bxneayn, yn+ = cxn[parenleftbig] eayn[parenrightbig].

Here, xn and yn represent the densities of the host and parasitoid population at year n. b is the number of ospring of an unparasitized host surviving to the next year. Assuming random encounter between hosts and parasitoids, the probability that a host escapes parasitism can be approximated by eayn, where a is a proportionality constant. Similarly, the probability to become infected is then given by eayn. The parameter c describes the number of parasitoids that hatch from an infected host.

Now, assume that the host has bounded dynamics in absence of parasitoid, i.e., has self-regulation (density dependence). For example, assume host dynamics are inherently logistic (e.g., the Beverton-Holt model).

Then a modied form of the Nicholson-Bailey host-parasitoid model is

xn+ = bxneayn

+ dxn , yn+ = cxn[parenleftbig] eayn[parenrightbig], ()

where a, b, c, d and the initial conditions x, y are positive real numbers.

In this paper our aim is to study the dynamics of system (). More precisely, we investigate the boundedness character, existence and uniqueness of a positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point, and the rate of convergence of positive solutions of system (). To investigate the dynamics we shall use standard results from theory of rational dierence equations. However, we shall state the results that we employ and refer the interested readers for a systematic study of rational dierence equations to [] and the references therein. In Refs. [] qualitative behavior of some biological models is discussed. The rest of the paper is organized as follows. In Section the required known results about linearized stability are given.

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Section discusses the boundedness character of the model. Section is about the existence and uniqueness of the positive equilibrium point. It also contains the local stability of the equilibrium point. Section discusses the global behavior of the equilibrium point. Whereas Section is about the rate of convergence and Section gives the numerical examples of the proved results. In the last section a brief conclusion is given.

2 Linearized stability

Let us consider the two-dimensional discrete dynamical system of the form

xn+ = f (xn, yn),

yn+ = g(xn, yn), n = , , . . . ,

(ii) An equilibrium point (x, y) is said to be unstable if it is not stable.

(iii) An equilibrium point (x, y) is said to be asymptotically stable if there exists >

such that (x, y) (x, y) < and (xn, yn) (x, y) as n .

(iv) An equilibrium point (x, y) is called a global attractor if (xn, yn) (x, y) as n .(v) An equilibrium point (x, y) is called an asymptotic global attractor if it is a global

attractor and stable.

Denition . Let (x, y) be an equilibrium point of the map F(x, y) = (f (x, y), g(x, y)), where f and g are continuously dierentiable functions at (x, y). The linearized system of ()

about the equilibrium point (x, y) is given by

Xn+ = F(Xn) = FJXn,

where Xn = [parenleftbig]

Let (x, y) be an equilibrium point of system (), then

x =

[parenrightBigg]

.

()

where f : I J I and g : I J J are continuously dierentiable functions and I, J are some intervals of real numbers. Furthermore, a solution {(xn, yn)}n= of system () is uniquely determined by initial conditions (x, y) I J. An equilibrium point of () is a point (x, y) that satises

x = f (x, y),

y = g(x, y).

Denition . Let (x, y) be an equilibrium point of system ().

(i) An equilibrium point (x, y) is said to be stable if for every > , there exists >

such that for every initial condition (x, y), (x, y) (x, y) < implies (xn, yn) (x, y) < for all n > , where is the usual Euclidian norm in

R.

xn

yn [parenrightbig] and FJ is a Jacobian matrix of system () about the equilibrium point (x, y).

, y = cx[parenleftbig] eay[parenrightbig]. ()

The Jacobian matrix of the linearized system of () about the xed point (x, y) is given by

FJ(x, y) =

[parenleftBigg] beay

(+dx)

abxeay+dx

c( eay) acxeay

bxeay

+ dx

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Lemma . [] Consider the second-degree polynomial equation

p q = , ()

where p and q are real numbers.(i) If both roots of Equation () lie in the open unit disk || < , then the equilibrium

point (x, y) is locally asymptotically stable.(ii) If at least one of the roots of Equation () has absolute value greater than one, then the equilibrium point (x, y) is unstable.

(iii) A necessary and sucient condition for both roots of Equation () to lie inside the open disk || < is

|p| < q < .

In this case the locally asymptotically stable equilibrium (x, y) is also called a sink.(iv) A necessary and sucient condition for both roots of Equation () to have absolute value greater than one is

|q| > , |p| < | q|.

In this case (x, y) is a repeller.(v) A necessary and sucient condition for one root of Equation () to have absolute value greater than one and for the other to have absolute value less than one is

p + q > , |p| > | q|.

In this case the unstable equilibrium (x, y) is called a saddle point.(vi) A necessary and sucient condition for a root of Equation () to have absolute value equal to one is

|p| = | q|.

In this case the equilibrium (x, y) is called a non-hyperbolic point.

3 Boundedness

The following theorem shows that every positive solution {(xn, yn)}n= of system () is bounded.

Theorem . Every positive solution {(xn, yn)}n= of system () is bounded.

Proof Let {(xn, xn)}n= be any positive solution of system (), then

xn+ = bxneayn

+ dxn

Also

bxn dxn

bd , n = , , . . . . ()

yn+ = cxn[parenleftbig] eayn[parenrightbig] cxn

bcd , n = , , . . . . ()

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Hence from () and (), we have

xn

bcd , n = , , . . . . ()

Theorem . Let {(xn, yn)} be a positive solution of system (). Then [, bd ] [, bcd] is an invariant set for system ().

Proof It follows from induction.

4 Existence and uniqueness of a positive equilibrium point and local stability

The following theorem shows the existence and uniqueness of a positive equilibrium point of system ().

Theorem . If b > and d < ac

b ln( +b

b ) , then system () has a unique positive equilibrium point (x, y) in [, bd ] [, bcd].

Proof Consider the following system:

x = bxeay + dx , y = cx[parenleftbig] eay[parenrightbig]. ()

Assume that (x, y) in [, bd ] [, bcd], then it follows from () that

y =

a

bd , yn

ln[parenleftbigg] b

[parenrightbigg], x = y

+ dx

c( eay).

Dene F(x) = h(x)

c(eah(x)) x, where h(x) = a ln(

b+dx ) and x [, bd ]. It is easy to see that

b ) . Hence, F(x) has at least one positive solution in the interval [, bd ]. Furthermore, it is easy to show that F (x) =

(eah(x)ah(x)eah(x))h (x)c(eah(x))c(eah(x)) < , where h (x) =

F() = b

ln bac(b) > if b > . Also, F( bd ) = bln(

+b b )

ac bd < if d <

ac

b ln( +b

da(+dx) for all x [, bd ]. Hence, F(x) =

has a unique positive solution x [, bd ].

Theorem . For the unique positive equilibrium point (x, y) in [, bd ] [, bcd] of system (), the following statements hold true:(i) The unique positive equilibrium point of system () is locally asymptotically stable if and only if

beacr(bdr+b)(acr( + bdr) + )

( + bdr)

< abcreacr(bdr+b)(bdr(eacr(bdr+b) ) )

( + bdr) < .

(ii) The unique positive equilibrium point is a repeller if and only if

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

abcreacr(bdr+b)(bdr(eacr(bdr+b) ) ) ( + bdr)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

>

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beacr(bdr+b)(acr( + bdr) + )

( + bdr)

< [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

(iii) The unique positive equilibrium point is a saddle point if and only if

[parenleftbigg]beacr(bdr+b)(acr( + bdr) + )

( + bdr)

+ [parenleftbigg]abcreacr(bdr+b)(bdr(eacr(bdr+b) ) )

( + bdr)

beacr(bdr+b)(acr( + bdr) + )

( + bdr)

> [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

(iv) The unique positive equilibrium point is non-hyperbolic if and only if

beacr(bdr+b)(acr( + bdr) + )

( + bdr)

= [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

Proof (i) The characteristic polynomial of the Jacobian matrix FJ(x, y) about the equilibrium point (x, y) is given by

P() = eay(acx( + dx) + b)

( + dx)

abcreacr(bdr+b)(bdr(eacr(bdr+b) ) )

( + bdr) .

and

abcreacr(bdr+b)(bdr(eacr(bdr+b) ) ) ( + bdr)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle].

[parenrightbigg] >

and

abcreacr(bdr+b)(bdr(eacr(bdr+b) ) ) ( + bdr)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle].

abcreacr(bdr+b)(bdr(eacr(bdr+b) ) ) ( + bdr)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle].

+ abcxeay(eay( + dx) dx)

( + dx)

. ()

As pointed out in [], it is convenient to discuss stability behavior in terms of the quantity r. So, for the equilibrium point (x, y) of system (), we have from system ()

eay = b + dr,

where r = xb is the ratio of steady-state x with b. Moreover,

y = cr(b bdr).

So, in terms of r, Equation () takes the form

P() = beacr(bdr+b)(acr( + bdr) + )

( + bdr)

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Let

p = beacr(bdr+b)(acr( + bdr) + ) ( + bdr) ,

q = abcreacr(bdr+b)(bdr(eacr(bdr+b) ) )

( + bdr) .

Then it follows from Lemma . that the unique positive equilibrium point (x, y) of system () is locally asymptotically stable if and only if

beacr(bdr+b)(acr( + bdr) + )

( + bdr) <

abcreacr(bdr+b)(bdr(eacr(bdr+b) ) )( + bdr) < .

Obviously, one can prove (ii), (iii) and (iv).

5 Global character

Lemma . [] Let I = [a, b] and J = [c, d] be real intervals, and let f : I J I and g : I J J be continuous functions. Consider system () with initial conditions (x, y) I J.

Suppose that the following statements are true:(i) f (x, y) is non-decreasing in x and non-increasing in y.(ii) g(x, y) is non-decreasing in both arguments.(iii) If (m, M, m, M) I J is a solution of the system

m = f (m, M), M = f (M, m),

m = g(m, m), M = g(M, M)

such that m = M and m = M, then there exists exactly one equilibrium point (x, y) of system () such that limn(xn, yn) = (x, y).

Theorem . Assume that ac + d > abc, then the unique positive equilibrium point (x, y) in [, bd ] [, bcd] of system () is a global attractor.

Proof Let f (x, y) = bxeay+dx and g(x, y) = cx( eay). Then it is easy to see that f (x, y) is non

decreasing in x and non-increasing in y. Moreover, g(x, y) is non-decreasing in both arguments x and y. Let (m, M, m, M) be a positive solution of the system

m = f (m, M), M = f (M, m),

m = g(m, m), M = g(M, M).

Then one has

m = bmeaM

+ dm , M =

and

bMeam

+ dM ()

m = cm[parenleftbig] eam[parenrightbig], M = cM[parenleftbig] eaM[parenrightbig]. ()

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From () one has

m = beaM

d , M =

beam

d . ()

From () one has

m = c(beaM )( eam)

d , M =

c(beam )( eaM)

d . ()

From () we have

m M = bd eamaM[parenleftbig]eam eaM[parenrightbig]. ()

From () we have

m M = c(b )d eamaM[parenleftbig]eam eaM[parenrightbig]. ()

Moreover, one has

eam eaM = ae (m M), ()

where

min{m, M} max{m, M}.

From () and (), we get

m M = abd eamaM+ (m M). ()

From () it follows that

|m M|

abd |m M|. ()

From () and (), we get

m M = ac(b )d eamaM+ (m M). ()

From () it follows that

|m M|

ac(b )d |m M|. ()

From () we get

[parenleftbigg]ac + d abc d

[parenrightbigg]

|m M| ,

from which it follows that m = M and similarly it is easy to show that m = M. Hence, from Lemma . the unique positive equilibrium point of system () is a global attractor.

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Lemma . The unique positive equilibrium point (x, y) in [, bd ] [, bcd] of system () is globally asymptotically stable if and only if

beacr(bdr+b)(acr( + bdr) + )

( + bdr) <

abcreacr(bdr+b)(bdr(eacr(bdr+b) ) )( + bdr) < .

Proof The proof is a direct consequence of Theorems . and ..

6 The rate of convergence

In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of system ().

The following result gives the rate of convergence of solutions of the system of dierence equations

Xn+ = [parenleftbig]A + B(n)[parenrightbig]Xn, ()

where Xn is an m-dimensional vector, A Cmm is a constant matrix, and B :

Z+

Cmm

is a matrix function satisfying

[vextenddouble][vextenddouble]B(n)[vextenddouble][vextenddouble] ()

as n , where denotes any matrix norm which is associated with the vector norm

[vextenddouble][vextenddouble](x, y)[vextenddouble][vextenddouble] = [radicalbig]x + y.

Proposition . (Perrons theorem []) Suppose that condition () holds. If Xn is a solution of (), then either Xn = for all large n or

= lim

n

[parenleftbig] Xn [parenrightbig]/n

exists and is equal to the modulus of one of the eigenvalues of matrix A.

Proposition . [] Suppose that condition () holds. If Xn is a solution of (), then either Xn = for all large n or

= lim

n

Xn+

Xn

exists and is equal to the modulus of one of the eigenvalues of matrix A.

Let {(xn, yn)} be any solution of system () such that limn xn = x and limn yn = y. To nd the error terms, one has from system ()

xn+ x =

bxneayn

+ dxn

bxeay

+ dx

= beayn

( + dxn)( + dx)

(xn x) +

bx(eayn eay) ( + dx)(yn y)

(yn y)

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and

yn+ y = cxn[parenleftbig] eayn[parenrightbig] cx[parenleftbig] eay[parenrightbig]

= c[parenleftbig] eayn[parenrightbig](xn x)

cx(eayn eay) yn y

(yn y).

Let en = xn x and en = yn y, then one has

en+ = anen + bnen

and

en+ = cnen + dnen,

where

an = beayn

( + dxn)( + dx)

, bn = bx(eayn eay) ( + dx)(yn y)

,

cn = c[parenleftbig] eayn[parenrightbig], dn = cx(eayn eay)

yn y

.

Moreover,

lim

n an =

beay

( + dx)

, lim

n bn =

abxeay + dx

,

n cn = c

lim

[parenleftbig] eay[parenrightbig], lim

n dn = acxeay.

Now the limiting system of error terms can be written as

[parenleftBigg]en+

en+

[parenleftBigg] beay

(+dx)

abxeay+dx

[parenrightBigg] = c( eay) acxeay

[parenrightBigg][parenleftBigg]en

en

[parenrightBigg]

,

which is similar to the linearized system of () about the equilibrium point (x, y).

Using proposition (.), one has following result.

Theorem . Assume that {(xn, yn)} is a positive solution of system () such that

limn xn = x, and limn yn = y, where x in [, bd ] and y in [, bcd]. Then the error vector en = [parenleftbig]

en

en [parenrightbig] of every solution of () satises both of the following asymptotic relations:

lim

n

= [vextendsingle][vextendsingle],FJ(x, y)[vextendsingle][vextendsingle],

where ,FJ(x, y) are the characteristic roots of the Jacobian matrix FJ(x, y).

7 Examples

In order to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent dierent types of qualitative behavior of solutions to the system of nonlinear dierence equations (). All plots in this section are drawn with Mathematica.

[parenleftbig] en [parenrightbig]

n = [vextendsingle][vextendsingle],FJ(x, y)[vextendsingle][vextendsingle], lim

n

en+

en

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(a) Plot of xn for system () (b) Plot of yn for system ()

(c) An attractor of system ()

Example Let a = ., b = ., c = ., d = .. Then system () can be written as

xn+ = .xne.yn

+ .xn , yn+ = .xn[parenleftbig] e.yn[parenrightbig], n = , , . . . , ()

with initial conditions x = ., y = ..

In this case the unique positive equilibrium point of system () is given by (x, y) = (., .). Moreover, in Figure the plot of xn is shown in Figure (a), the plot of yn is shown in Figure (b) and an attractor of system () is shown in Figure (c).

Example Let a = ., b = ., c = ., d = .. Then system () can be written as

xn+ = .xne.yn

+ .xn , yn+ = .xn[parenleftbig] e.yn[parenrightbig], n = , , . . . , ()

with initial conditions x = ., y = ..

In this case the unique positive equilibrium point of system () is given by (x, y) = (., .). Moreover, in Figure the plot of xn is shown in Figure (a), the plot of yn is shown in Figure (b) and an attractor of system () is shown in Figure (c).

Example Let a = ., b = ., c = ., d = .. Then system () can be written as

xn+ = .xne.yn

+ .xn , yn+ = .xn[parenleftbig] e.yn[parenrightbig], n = , , . . . , ()

with initial conditions x = ., y = ..

Figure 1 Plots for system (23).

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(a) Plot of xn for system () (b) Plot of yn for system ()

(c) An attractor of system ()

In this case the unique positive equilibrium point of system () is given by (x, y) = (., .). Moreover, in Figure the plot of xn is shown in Figure (a), the plot of yn is shown in Figure (b) and an attractor of system () is shown in Figure (c).

Example Let a = ., b = ., c = ., d = .. Then system () can be written as

xn+ = .xne.yn

+ .xn , yn+ = .xn[parenleftbig] e.yn[parenrightbig], n = , , . . . , ()

with initial conditions x = ., y = ..

In this case the unique positive equilibrium point of system () is unstable. Moreover, in Figure the plot of xn is shown in Figure (a), the plot of yn is shown in Figure (b) and a phase portrait of system () is shown in Figure (c).

Example Let a = ., b = ., c = ., d = .. Then system () can be written as

xn+ = .xne.yn

+ .xn , yn+ = .xn[parenleftbig] e.yn[parenrightbig], n = , , . . . , ()

with initial conditions x = ., y = ..

In this case the unique positive equilibrium point of system () is unstable. Moreover, in Figure the plot of xn is shown in Figure (a), the plot of yn is shown in Figure (b) and a phase portrait of system () is shown in Figure (c).

Figure 2 Plots for system (24).

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(a) Plot of xn for system () (b) Plot of yn for system ()

(c) An attractor of system ()

Figure 3 Plots for system (25).

(a) Plot of xn for system () (b) Plot of yn for system ()

(c) Phase portrait of system ()

Figure 4 Plots for system (26).

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(a) Plot of xn for system () (b) Plot of yn for system ()

(c) Phase portrait of system ()

8 Conclusion

This work is related to the qualitative behavior of the modied Nicholson-Bailey host-parasitoid model. We have investigated the existence and uniqueness of positive steady-state of system (). Under certain parametric conditions, the boundedness of positive solutions is proved. Moreover, we have shown that the unique positive equilibrium (x, y)

in the [, bd ] [, bcd] point of system () is locally asymptotically stable if and only if

beacr(bdr+b)(acr(+bdr)+)

(+bdr) < ab

creacr(bdr+b)(bdr(eacr(bdr+b)))

(+bdr) < hold true. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. Furthermore, the rate of convergence of positive solutions of () which converge to its unique positive equilibrium point is demonstrated. Finally, some numerical examples are provided to support our theoretical results. These examples are experimental verication of our theoretical discussions.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Acknowledgements

The authors thank the main editor and anonymous referees for their valuable comments and suggestions that led to the improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.

Received: 3 October 2014 Accepted: 2 January 2015

References

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Figure 5 Plots for system (27).

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