Delayed prey-predator system with habitat complexity ...

International Conference on Mathematical Sciences

Delayed prey-predator system with habitat complexity and refuge Shashi Kant, Vivek Kumar Department of Applied Mathematics, Delhi Technological University,Formerly Delhi College of Engineering, Main Bawana Road, Shahbad Daulatpur, Delhi-110042, India. [email protected]

______________________________________________________________________________________________________ Abstract In this paper, we propose and study a prey-predator model with habitat complexity and prey refuge. Firstly, we modify the original Holling type II functional response and propose a new type of functional response of the form . Secondly, we observe the local stability of coexistence equilibrium and investigate condition for Hopf-bifurcation. Thirdly, formulae are derived for the direction and stability of Hopfbifurcation. Finally, to validate our theoretical framework, a numerical example is considered. Keywords: Prey-predator system, Time delay, Stability analysis, Hopf-bifurcation.

______________________________________________________________________________________________________ 1. Introduction Prey-predator study is an important branch of applied mathematics (see, for example [1-4]). Functional response is an important aspect in prey-predator dynamics. Many functional responses have been proposed: Holling type, ratiodependent Michaelis-Menten type, Ivlev type, Crowley-Martin type, Beddington-DeAgelis type, Hassell-Varley type etc. [ 5-13]. Classical prey-predator model with logistic growth takes the form:

where x and y denotes the prey and predator populations respectively. d denotes the mortality of predator population and θ is conversion efficiency. p(x) is functional response, the choices may be any of the responses as mentioned above. For example, if we take , the model (1) takes the form:

where c is the dimensionless parameter and measures the degree or strength of habitat complexity. Further c is also satisfied the condition 0 < c < 1. If c = 0, then it means there is no habitat complexity and above equation reduced to original response in equation (4). Thus, if we include the case of c = 0 in the assumptions of [16], we get the range for the constant c as 0 ≤ c < 1. In [16] using modified type II response (5), N. Bairagi and D. Jana proposed a model

Introducing the gestation delay form

model (6) takes the

(7) Holling type II functional response is the most commonly used Further, if predator faces difficulties in attacking and catching response and is given by the prey, for example, if prey is stronger then predator or predator is facing disease or the case of juvenile predator, then where is maximum growth rate and is the half saturation we can introduce one more constant say β in the functional constant. This is also written as response (5) of N. Bairagi and D. Jana [ 16], we get a new type of functional response: where is attack coefficient and is the handling time. N. Bairagi and D. Jana [16] introduced the habitat complexity and proposed the following Holling type II functional Normally and the case with will response of the form: give the functional response (5) as discussed in [16]. If 584

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then we have the original Holling type II functional response (4). We call as a difficulty coefficient. Now, we also assumed that there is a refuge protecting of prey, where is constant. This leaves of prey available to the predator. For references on refuge we can refer [14, 15]. By incorporating these assumptions, the model (7) takes the form

(9)

The characteristics equation may be written as

where etc. are positive constants. Meanings are given in the Table 1. Table 1. Biological/Ecological Meaning of the Symbols

Symbol

Biological Meaning Prey density Predator density Intrinsic growth rate Carrying capacity Conservation efficiency Attack coefficient Refuge Handling time Degree of habitat complexity Difficulty coefficient Death rate of predator

Numerical Range Variable Variable Any positive value Any positive value Any positive value Positive value

When there is no delay , the corresponding characteristics equation of the system (9) is given by i.e.

Any positive value

2. Stability Analysis System (9) admits three equilibrium points (i) trivial equilibrium point (ii) boundary equilibrium in this situation predator population will die out (iii) non zero (interior) equilibrium point. Non zero equilibrium point provide the coexistence of prey and predator both.

roots of this equation are given by Thus for the stability of the non zero equilibrium of the system (9) without delay, the roots must have negative real parts. Now for delayed system, to determine that the interior equilibrium point we start with the assumption that the system without delay has stable interior equilibrium point. Let we seek purely imaginary roots of We have, equating real and imaginary parts, we have

We shall focus on the stability of the non zero equilibrium point. Any member of the non zero equilibrium point must satisfy squaring and adding, we have or

Let The system (9) can be written in matrix form after linearization process: We see that

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always. Now two cases arise (i) If , then above Eq. Will have roots with negative real parts for all therefore the interior equilibrium will be locally asymptotically stable. (ii) If , then above Eq. Has one positive root. In other words Eq. (18) will have a positive root (say) and therefore characteristics equation (12) will have a pair of purely imaginary roots for some delay i.e. Eq. (16)-(17) will give

(22) By Riesz representation theorem, there exists a bounded variation function for such that

Differentiating Eq. (12 ) with respect to , it is easy to prove the transversality condition is satisfied.

We can choose

Indeed

where is Dirac delta function. For us define two functions:

. Now we can state the main

result of this section in the following theorem: Theorem 1. Suppose the system (9) is locally asymptotically stable without delay around the interior equilibrium then the following results hold: (i) If then the interior equilibrium is locally asymptotically stable for and unstable when where

, (24) let

(25) and (26) Then FDE system is equivalent to (27)

(ii) When a Hopf-bifurcation occurs as passes through the critical value (iii) If then the interior equilibrium is locally asymptotically stable for all

where For

for define

.

(28) and also a bilinear product

3. Direction and Stability of Hopf-bifurcation In recent years the calculation of direction and stability of Hopf-bifurcation have been seen in many papers [3, 16]. In where Then, and are adjoint operators. previous section, we see that Hopf-bifurcation occurs at a Suppose that and are eigen vectors of and critical value of delay corresponding to the eigen values and respectively. Then suppose that is the eigen vector of corresponding to the eigen value , then = . In this section, we will also follow the procedure given by Hassard et al [17]. It is assumed that, our system undergoes Hopf-bifurcation at a critical value and are the corresponding values of the characteristics equation (12). Let then is the Hopf- where bifurcation value of the system. Normalizing the time delay by we can write the system in the form of a functional differential equation (FDE) in Banach space where where

and and

as

thus

where

are given, respectively, by

are defined in previous section and

. Similarly, let

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or

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International Conference on Mathematical Sciences

Since, value of

and

are adjoint, therefore we have, from this equation, we can compute the

. ,

hence, Define

on the center manifold . The value of following series expansion:

is given by

Since as computed above has ( ) and ( ), we need to compute them. From Eq. (27) and (30), we have, where and are local coordinates for the center manifold in the direction of and . Note that manifold is real if is real. We only consider real solutions. For solution of (20), since we have, = where

(say). Thus

=

Remark 1. The corresponding equation (4.14) (Page 3261) in [16] is written as =

By Eq. (30)-(31), we have which has a typing error and should be corrected as =

Therefore, we have = and where value of

Expanded version of is obtained after putting the values of the terms etc. within the above Eq. And the same is listed at Appendix at the end of the paper. From that equation and Eq. (32), by comparing the coefficients of same powers of , we get the coefficients viz.

should be corrected as is given as (equation 4.15 in [16])

should be corrected as

Rewrite the Eq. (34) as ,

(35) where

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and

Thus we obtain, where

By Eq. (34) and (36), we have

(38) By using Eq. (34), for

+2

, ,

(45) +2

. Remark 2. The equation (4.17) in [16] is “we know that for ,

Substituting (41) and (42) into (43) and note that )

has a typing

and )

error and it should be corrected as “we know that for

(46)

we obtain

d

,

)

2

,

Now compare the coefficients, we have the following

(40)

, where

From Eq. (34) and (40) and the definition of A, it follows that ( )

Since Remark 3. The corresponding equation in [16] at page (3262) is

, we have + where obtain

, (41)

is a constant vector. Similarly, we

+

,

(42)

has a typing error and it should be corrected as

where is a constant vector. The vectors and are required to be determined in the following steps. From the definition of and Eq. (38), we obtain

And

is given by 588

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. Therefore the components of vectors

and

Theorem 2. (i) determines bifurcation. If , supercritical (subcritical) and solutions exist for (

are given by

the direction of the Hopfthen the Hopf-bifurcation is the bifurcating periodic ).

(ii) determines the stability of the bifurcating solutions: the bifurcating periodic solutions are stable (unstable) if

(iii) determines the period of the bifurcating periodic solutions: the period increases (decreases) if

4. Numerical Example

,

We

select

a

hypothetical

set

of

parameters

It is observed that non zero equilibrium exists and is locally stable. It is calculated that is negative and , hence by Th. 1 the Hopf-bifurcation occurs as the system passes through Few graphs are drawn in Fig. 1 by using ode 45 of MATLAB for without delay. Similar graphs for delayed model with are drawn in Fig. 2 by dde23 of MATLAB.

where

,

5. Discussion In this paper, we modified the model of N. Bairagi and D. Jana [16] by including refuge and introducing new functional response of the form

Thus we can determine the values of by using the values of the vectors and as determined above. Hence, by , is completely determined. Hence, we can compute the following values:

We analyze the

model in terms of local stability. By using the central manifold reduction, we derive the formulae for direction and stability of Hopf-bifurcation. Theorem 1 and 2 are the main results of the paper. By assuming a hypothetical set of parameters we solve the model by ode45 and dde23 of MATLAB. It is also interesting to note that we list three remarks which provide the errors in the study of N. Bairagi and D. Jana [16]. It is also remarkable that real parameters are not available for this model. As an application, this model can be used to study real ecosystems. Real parameters may be investigated.

References. .

[1] B. Mukhopadhyaya and B. Bhattacharyya, Dynamics of a delay-diffusion prey-predator Model with disease in the prey, J. Appl. Math and Comp. 17(12) (2005), 361 – 377. http://dx.doi.org/10.1007/ BF02936062.

Now we can state the main theorem of this section:

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International Conference on Mathematical Sciences [2] Soovoojeet Jana and T.K. Kar, Modeling and Analysis of a prey-predator system with disease in the prey Chaos, Solitons and Fractals, 47(2013), 42– 53. http://dx.doi.org/10.1016/j.chaos.2012.12.002.

6. Appendix

[3] Guang -Ping Hu and Xiao-Ling Li, Stability and Hopf bifurcation for a delayed predator-prey model with disease in the prey, Chaos, Solitons and Fractals , 45 (2012), 229 – 237. http://dx.doi.org/10.1016/j.chaos.2011.11.011. [4] D. Greenhalgh and M. Haque, A predator-prey model with disease in the prey species only, Math. Methods Appl. Sci., 30(8) (2007), 911 – 929. http://dx.doi.org/10.1002/mma.815. [5] C.S. Holling, The components of predation as revelaed by a study of small mamal predation of the European pine sawfly, Canad. Entomologist, 91(1959), 293–320. [6] C.S. Holling, Some chararacteristics of simple types of predation and parasitism, Canad. Entomologist, 91(1959), 385–395. [7] Dawes JH and Souza MO, A derivativation of Holling’s type I,II and III functional responses in predator-prey system, J. Theor.Biol.,327(2013),11– 22. http://dx.doi.org/10.1016/j.thbi.2013.02.017

+.....

[8] D.L. Deangenis et al, A model for trophic interaction, Ecology, 56 (1975), 881–892. [9] Ivlev,V.S., Experiment Ecology of the feeding of fishes, Yale University Press, New Haven, C.T.1961. [10] J.R. Beddington, Mutual interference between parasites or predators and its eff ect on searching efficiency, J. Animal Ecology, 44 (1975), 331–340. [11] M.P. Hassell and C.C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133– 1137. [12] P.H. Crowley and E.K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. North American Benthological Soc., 8(1989), 211–221. [13] R. Arditi and L.R. Ginzburg, Coupling in predator-prey dynamics: Ratiodependence, J. Theor. Bio., 139(1989), 311–326. [14] Tapan Kumar Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Communications in Nonlinear Science and Numerical Simulation, 10(2005), 681–691. http://dx.doi.org/10.1016/j.cnsns.2003.08.006. [15] Sapna Devi, Effects of prey refuge on a ratio-dependent predator-prey model with stage structure of prey population, Appl. Math. Model., 37(2013), 4337-4349. http://dx.doi.org/10.1016/j.apm.2012.09.045. [16] N. Bairagi and D. Jana, On the stability and Hopf-bifurcation of a delayinduced predator-prey system with habitat complexity, Appl. Math. Model., 35(2011), 3255-3267. http://dx.doi.org/10.1016/j.apm.2011.01.025. [17] B.D Hassard et al, Theory and Application of Hopf bifurcation, Cambridge University, Cambridge, 1981.

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Figure 2: Solution of (9) with delay ( for initial conditions x(0)=0.9, y(0)=0.9.

) with parameter set

Figure 1: Solution of (9) without delay (

) with parameter set

for initial conditions x(0)=0.9, y(0)=0.9, the positive equilibrium is stable.

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