The spatial patterns through diffusion-driven instability

Applied Mathematical Modelling 36 (2012) 1825–1841

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The spatial patterns through diffusion-driven instability in a predator–prey model Lakshmi Narayan Guin a, Mainul Haque b,⇑, Prashanta Kumar Mandal a a b

Department of Mathematics, Visva-Bharati, Santiniketan 731 235, West Bengal, India School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

a r t i c l e

i n f o

Article history: Received 6 December 2010 Received in revised form 28 April 2011 Accepted 23 May 2011 Available online 19 July 2011 Keywords: Diffusion-driven instability Hopf-bifurcation Predator–prey model Reaction–diffusion system Self and cross-diffusion Simulation

a b s t r a c t Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator–prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction In 1952, Alan Turing first proposed the reaction–diffusion theory for pattern formation in his seminal work on the chemical basis of morphogenesis [1]. A fact in which a non-linear system is asymptotically stable in the absence of self and crossdiffusion but unstable in the presence of self and cross-diffusion is known as Turing instability. This concept has been playing significant roles in theoretical ecology, embryology and other branches of science [2,3]. In the history of population ecology, the stability behaviour of a system of interacting populations by taking into account the effect of self as well as cross-diffusion in the prey-dependent predator–prey models has received much attention by both ecologists and mathematicians [4– 6]. The term self-diffusion which implies the per capita diffusion rate of each species is influenced only by its own density, i.e. there is no response to the density of the other one. On the other hand, cross-diffusion implies the per capita diffusion rate of each species which is influenced by the other ones. The value of the cross-diffusion coefficient may be positive, negative or zero. The positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species while the negative cross-diffusion coefficient for one species tends to diffuse in the direction of higher concentration of another species. There are several popular predator–prey models available in the current literature. In this investigation, we consider a prey-dependent predator–prey model in which predator has alternative source of food. The predator has a logistic growth law in case of extinction of the prey as predator has alternate source of food other than the prey available to it (humans, leopards and dogs). Similar types of creative research works in this domain of interest are available in the studies of van Baalen ⇑ Corresponding author. E-mail addresses: [email protected] (L.N. Guin), [email protected], [email protected] (M. Haque), [email protected] (P.K. Mandal). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.055

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et al. [7], Petrovskii and Malchow [8], Morozov et al. [9] and Haque and Greenhalgh [10]. By incorporating the alternative food source in the general prey-dependent predator–prey model [7–10], we obtain

  du u puv  ¼ au 1  ¼ f1 ðu; v Þ; dt k1 mu þ n   dv v epuv þ ¼ bv 1   dv ¼ f2 ðu; v Þ; dt k2 mu þ n uð0Þ ¼ U 0 P 0;

ð1:1aÞ ð1:1bÞ

v ð0Þ ¼ V 0 P 0;

ð1:1cÞ

where u denotes the biomass of the prey population and v that of the predator population. The parameters a and k1 designate the intrinsic growth rate, carrying capacity of the prey species, respectively; while b and k2 are the corresponding parameters to the predator. p denotes predation rate, e, the conversion factor and d represents the predator death rate. All parameters of the system are naturally positive including m and n, where both of them represents half-saturation constants. Here we assume that the predator has a logistic growth rate since it has sufficient resources for alternative food and it may be argued that alternative food sources may have an important role in promoting the persistence of predator–prey systems [7]. The switching made by the predator species has a significant contribution to the persistence of species in the predator–prey system. The theory of spatial patterns formation in predator–prey system via Turing instabilities plays an important role in ecology like other non-linear systems naturally arise in many practical problems of science, engineering and technology. Modelling reaction–diffusion equations has become an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal prey dynamics, fractal properties of predator movements and their interrelationships [11]. Assuming the importance of self and cross-diffusion on predator–prey model, an attempt is made to investigate the role of self as well as cross diffusion in a prey-dependent predator–prey model where predator has an alternative source of food.

2. Mathematical model and its analysis In order to show the effect of self and cross-diffusion to the system (1.1a)–(1.1c), we use the following set of differential equations

@u ¼ f1 ðu; v Þ þ D11 r2 u þ D12 r2 v ; @t @v ¼ f2 ðu; v Þ þ D21 r2 u þ D22 r2 v ; @t uð0; nÞ ¼ U 0 ðnÞ > 0; v ð0; gÞ ¼ V 0 ðgÞ > 0;

ð2:1aÞ ð2:1bÞ ð2:1cÞ

where

r2 

@2 2

@n

þ

@2 @ g2

represents the usual Laplacian operator in two dimensional space of real variables n and g in Cartesian coordinates. Here the environment is assumed to be uniform, that is, the system parameters do not depend on space or time. The parameters D22, D11 are the positive self-diffusion coefficients while D12, D21 are the cross-diffusion coefficients of the predator and the prey ~ ¼ ku ; v ~ ¼ kv ; ~t ¼ ta; ~ ~ ð0; nÞ ¼ U 0kðnÞ ¼ u0 ðnÞ; v ~ ð0; gÞ ¼ V 0kðgÞ ¼ v 0 ðgÞ, we get the follow~ ¼ gL ; u respectively. Assuming u n ¼ nL ; g 1 2 1 2 ing set of partial differential equations (after dropping tildes)

@u ¼ F 1 ðu; v Þ þ d11 r2 u þ d12 r2 v ; @t @v ¼ F 2 ðu; v Þ þ d21 r2 u þ d22 r2 v ; @t uð0; nÞ ¼ u0 ðnÞ > 0; v ð0; gÞ ¼ v 0 ðgÞ > 0; F 1 ðu; v Þ ¼ uð1  uÞ 

where pk  ¼ amk

2 1

uv  ; uþa

ð2:2aÞ ð2:2bÞ ð2:2cÞ

F 2 ðu; v Þ ¼ bv ð1  v Þ þ

ep ; b ¼ ba ; c ¼ am ; d ¼ da ; d11 ¼ DaL112 ; d12 ¼ DaL122

k2 k1

uv c uþa

 dv

d21 ¼ DaL212

k1 k2

and

the

dimensionless

parameters

are

a ¼ mkn 1 ;

and d22 ¼ DaL222 ; L being the characteristic length.

The system (2.2a)–(2.2c) has following biologically meaningful boundary equilibrium points: (i) E0(0, 0) (total extinction), (ii) E1(1, 0) (extinction of the predator), (iii) E2 ð0; 1  bdÞ (extinction of the prey) and (iv) one positive interior equilibrium 3 þaÞ point E3(u3, v3) (coexistence of prey and predator) where v 3 ¼ ð1u3 Þðu and u3 be the positive root of the equation 

bu3 þ ð2ab  bÞu2 þ ðb  2ab þ c  d þ a2 bÞu þ ðab  a2 b  adÞ ¼ 0: The conditions of the positiveness of the root u3 are given in Appendix A.

ð2:3Þ

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3. Stability analysis In this section, we deal with the stability and bifurcation analyses of the system defined by (2.2a)–(2.2c). The Jacobian matrix J of the system is given by

" J¼

v a 1  2u  ðuþ aÞ2

u uþa

acv

uc bð1  2v Þ þ uþ ad

ðuþaÞ2

# ¼ ðCij Þ22 :

Let us introduce the following notations in the system under consideration so that it can be written in the form X_ = F(X, b) = (F1(u, v), F2(u, v))T where X = (u, v)T and the Jacobian matrix of the system J  DF(X, b). We denote Jk = J evaluated at Ek, k = 0, 1, 2, 3 and the determinant Jk = det Jk, trace Jk=tr(Jk).   3.1. The behaviour of the system around the equilibrium points E0(0, 0), E1(1, 0) and E2 0; 1  bd We summarise the behaviours of the system (2.2a)–(2.2c) around the equilibrium points E0, E1 and E2 without proof. Proofs are straight forward and interested readers are referred to Haque and Venturino [12]. Lemma 1. (i) The equilibrium point E0(0, 0) is unstable for b > d and it becomes a saddle point if b < d. (ii) The system enters into a transcritical bifurcation around the equilibrium point E0(0, 0) at b = b[tc], where b[tc] = d. Lemma 2. (i) The equilibrium point E1(1, 0) is locally asymptotically stable iff b þ 1þc a  d < 0. (ii) E1(1, 0) is globally asymptotically stable if g11 > 0, g22 > 0 and N1 g 11 ¼ 12 ; g 22 ¼ 2ð1aþbþb aþcddaÞ ; g 12 ¼ g 21 ¼ 2D , 1

g11g22  g12g21 > 0

[13,14]

where

N1 ¼ 1 þ d þ 2a  b  c  2ba þ 2da  ba þ da  b  c þ d þ a2  ba2  ac þ da2 ; and D1 ¼ ð1 þ a  b  ba  c þ d þ daÞ; (iii) The system enters into a transcritical bifurcation around the equilibrium point E1(1, 0) at c = c[tc], where c[tc] = (d  b)(1 + a).

Lemma 3.     (i) The point E2 0; 1  bd is locally asymptotically stable iff d < b 1  a ; d < b and 0 < a < .  equilibrium  (ii) E2 0; 1  bd is globally asymptotically stable if h11 > 0, h22 > 0 and h11h22  h12h21 > 0 [13,14] where h11 ¼

N2 c 1 ; N 2 ¼ c2 b  c2 d  b2 a2 þ b2 a  bad  b2 a2 d þ b3 a2 ; h22 ¼ ð2b  2dÞ ; h12 ¼ h21 ¼ 2D2 2ðba þ b  d  dba þ b2 aÞ

and D2 ¼ b2 a2  2b2 a þ 2bad þ b2 a2 d  b3 a2 þ 2 b2  22 bd  2b2 da þ b3 a þ 2 d2 þ d2 ba:

  ½sn ½sn (iii) The system enters into a saddle-node bifurcation at around the equilibrium point E2 0; 1  bd b ¼ b1 ; b2 , where ½sn ½sn d b1 ¼ d and b2 ¼ x. 3.2. The behaviour of the system around the equilibrium point E3(u3, v3)

Lemma 4. (i) The equilibrium point E3(u3, v3) is locally asymptotically stable iff

2u33 þ u23 ð4a þ 2bv 3 þ d  1  b  cÞ þ u3 ð2a2 þ 4abv 3 þ 2ad  2a  2ab  acÞ þ ðv 3 a þ 2a2 bv 3 þ a2 d  a2  a2 bÞ >0 and



 u23 þ 2u3 a þ a2  2u33  4u23 a  2u3 a2  v 3 a ðu3 b þ ab  2u3 v 3 b  2v 3 ab þ u3 c  u3 d  adÞ þ u3 v 3 ac > 0:

(ii) E3(u3, v3) is globally asymptotically stable if k11 > 0, k22 > 0 and k11k22  k12k21 > 0 where the values of kij (i, j = 1, 2) are obtained by solving the equation (B.3) in Appendix B. (iii) The system passes through a saddle-node bifurcation around E3(u3, v3) at  = [sn], where [sn] satisfies the equality

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½sn ¼

 3 1 4u3 bv 3  2u33 b  2u33 c þ 2u33 d þ 8u23 bv 3 a þ 4u23 da  2u23 bv 3 þ u23 c  2u23 ac  4u23 ba  u23 d ðb þ 2v 3 b þ dÞv 3 a  þ u23 b  2u3 a2 b þ 4u3 a2 bv 3  4v 3 abu3 þ 2abu3  2adu3 þ u3 ac þ 2u3 a2 d þ a2 b  2v 3 a2 b  a2 d :

(iv) The system enters into a Hopf-bifurcation around E3(u3, v3) at

 = [hb], where [hb] satisfies the equality



u c ðu þ aÞ2 ½hb ¼ 1  2u3 þ bð1  2v 3 Þ þ 3  d 3 : u3 þ a v 3a

Proof. (i) The eigenvalues of the corresponding Jacobian matrix J at E3 (u3, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i k1 ¼ 1  2u3  ðuv 3þaaÞ2 þ bð1  2v 3 Þ þ uu33þca  d ; k2 ¼ k21  4k3 and k3 ¼ det J 3 .

v3)

are given by

1 ðk1 2

 k2 Þ, where

3

Thus the interior equilibrium point E3(u3,v3) is locally asymptotically stable iff k1 < 0 and k3 > 0. (ii) See Appendix B for proof and Fig. 1 shows a global stability situation of the system around E3. (iii) One of eigenvalues of J3 will be zero iff det J3 = (C11C22  C21C12) = 0 which gives  = [sn]. The other eigenvalue is evaluated by solving tr (J3) = C11 + C22 at [sn] and should be negative in order to get a saddlenode bifurcation [15]. Let h and w are the eigenvectors corresponding to the eigenvalue 0 of the matrices J3 and J T3 , h2 respectively. We obtain that h = (h1, h2)T and w = (w1, w2)T, where h1 ¼ hC211C12 ¼ CC22 ; w1 ¼ wC211C21 ¼ CC2212w2 ; h2 and w2 21

are any two non-zero real numbers. Now wT ½F  ðX 3 ; ½sn Þ ¼ uu33vþ3aw1 – 0 and wT[D2F(X3,[sn])(h, h)] – 0 where

X3 = (u3, v3). So, the system experiences a saddle-node bifurcation [15] around E3(u3, v3) at  = [sn]. (iv) The characteristic equation corresponding to the equilibrium point E3(u3, v3) is given by l211  k1 l11 þ k3 ¼ 0, where we assume ~ u  el11 t ; ~ v  el11 t . If k1 = 0 i.e. tr(J3) = 0, then both the eigenvalues will be purely imaginary provided det J3(=k3) is positive and there are no other eigenvalues with negative real part. Now k1 = 0 gives  ¼ e½hb . Substituting l11 = p1 + iq1 into the equation

 2  p1  q21  k1 p1 þ k3 ¼ 0;

l211  k1 l11 þ k3 ¼ 0 and separating real and imaginary parts we obtain ð3:1aÞ

2p1 q1  k1 q1 ¼ 0:

ð3:1bÞ

Differentiating (3.1b) both sides with respect to

 and considering p1 = 0, we get



dp1 v 3 a ¼ – 0: d ¼½hb 2ðu3 þ aÞ2 For a change of stability about E3, we should have the real part of l11 i.e. p1 = 0. Hence, the system undergoes a Hopf-bifurcation at E3 as  passes through the value [hb] Fig. 2 illustrates a Hopf-bifurcation situation of the system around E3. The parameter values are given in the figure. h

4. Stability analysis with diffusion The characteristic equation corresponding to the equilibrium point E3(u3, v3) is given by

l2  ðC11 þ C22 Þl þ ðC11 C22  C12 C21 Þ ¼ 0

ð4:1Þ

~ where we assume ~ u v Turing condition is one in which the uniform steady state of a RDE (Reaction–Diffusion Equation) is stable for the ODE, but it is unstable for the corresponding PDE with diffusion terms [16]. Now the conditions for the uniform steady state to be stable for the ODE are given by elt ;

elt .

trðJ 3 Þ ¼ ðC11 þ C22 Þ < 0;

ð4:2Þ

det J 3 ¼ ðC11 C22  C12 C21 Þ > 0;

ð4:3Þ

where J3 is the Jacobian matrix of the system evaluted at E3. Next we consider the Turing instability of the positive steady state E3(u3, v3) of the present model. To linearise the system around E3(u3, v3) for space and time perturbation, we assume

ðr; tÞ and v ð~  ðr; tÞ  u3 ; v ðr; tÞ  v 3 uð~ r; tÞ ¼ u3 þ u r; tÞ ¼ v 3 þ v ðr; tÞ; where u   ðr; tÞ a1 lt i~k~r u v ðr; tÞ ¼ a2 e e , where l is the growth rate of perturbation in time t; a1, a2 being the corresponding amplitudes and k, the wave-number of the solution. The characteristic equation of the system is

and r is spatial vector in two-dimensions. We start by letting the solution of the form



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2

2

2

2

ðC11  k d11  lÞðC22  k d22  lÞ  ðC12  k d12 ÞðC21  k d21 Þ ¼ 0; i:e: l ¼ 4

p B  ðB2  4CÞ 2 ; where B ¼ k ðd11 þ d22 Þ  ðC11 þ C22 Þ; 2

2

C ¼ det J 3 þ k ðd11 d22  d12 d21 Þ  k ðd11 C22 þ d22 C11  d12 C21  d21 C12 Þ: Now we need to identify conditions under which real parts of l are negative. If this condition is satisfied for all k, the stationary homogeneous state of the RDE is stable to all perturbations. 4.1. The roots of the dispersion relation Now we are, however, interested to find the conditions under which stationary state is not stable to spatial perturbations with k – 0. Consider two roots l1, l2 of the dispersion relation l = l(k2) where (l1 + l2) = B and l1l2 = C. Since tr(J3) < 0, B is positive for all k and hence both the roots can not be positive. Therefore, instability can only be achieved in case of different signs: l1 > 0, l2 < 0 or l2 > 0, l1 < 0. This can happen only if C(k2) < 0. To satisfy the condition C(k2) < 0, the coefficient of k2 must be positive i.e.

d11 C22 þ d22 C11  d12 C21  d21 C12 > 0:

ð4:4Þ 2

This is necessary, however, not a sufficient condition and we must require that the minimum of C(k ) is below zero. Now 2

kmin ¼

d11 C22 þ d22 C11  d12 C21  d21 C12 : 2ðd11 d22  d12 d21 Þ 2

2

Therefore the value of C(k2) at k ¼ kmin is

ðd11 C22 þ d22 C11  d12 C21  d21 C12 Þ2 > det J 3 4ðd11 d22  d12 d21 Þ

ð4:5Þ

i.e.

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      d22 d12 d21 d22 d12 d21  C21 >2 ðdet J 3 Þ: C22 þ C11 þ C12  d11 d11 d11 d11 d11 d11



Here (4.4) and (4.5) are the conditions under which stationary state is not stable to spatial perturbations with k – 0. 4.2. Turing’s diffusion-driven instability Finally, we can assemble all the results obtained from the entire analysis as follows: (i) tr(J) = (C11 + C22) < 0. i.e.

2u33 þ u23 ð4a þ 2bv 3 þ d  1  b  cÞ þ u3 ð2a2 þ 4abv 3 þ 2ad  2a  2ab  acÞ þ ðv 3 a þ 2a2 bv 3 þ a2 d  a2  a2 bÞ > 0: (ii) det J = (C11C22  C12C21) > 0. i.e.



 u23 þ 2u3 a þ a2  2u33  4u23 a  2u3 a2  v 3 a ðu3 b þ ab  2u3 v 3 b  2v 3 ab þ u3 c  u3 d  adÞ þ u3 v 3 ac > 0:

(iii) (d11C22 + d22C11  d12C21  d21C12) > 0. i.e.

 2 u3 ðb þ c  dÞ þ u3 ð2ab þ ac  2adÞ  2u23 v 3 b  4u3 v 3 ab  2v 3 a2 b þ a2 ðb  dÞ þ (iv)

d12  d22  2 d21  2 u ð1  4aÞ þ 2u3 að1  aÞ  2u33  v 3 a þ a2  ðv 3 acÞ þ u  þ u3 a > 0: d11 3 d11 d11 3

ðd11 C22 þd22 C11 d12 C21 d21 C12 Þ2 4ðd11 d22 d12 d21 Þ

> det J 3 .i.e.

 2 u3 ðb þ c  dÞ þ u3 ð2ab þ ac  2adÞ  2u23 v 3 b  4u3 v 3 ab  2v 3 a2 b þ a2 ðb  dÞ d12  d22  2 d21  2 u3 ð1  4aÞ þ 2u3 að1  aÞ  2u33  v 3 a þ a2  ðv 3 acÞ þ u3  þ u3 a þ d11 d11 d11 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s d22 d12 d21  K; > 2ðu3 þ aÞ d11 d11 d11

where

  K ¼ u23 þ 2u3 a þ a2  2u33  4u23 a  2u3 a2  v 3 a ðu3 b þ ab  2u3 v 3 b  2v 3 ab þ u3 c  u3 d  adÞ þ u3 v 3 ac:

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Satisfaction of all four of the above conditions guaranteesthat the  spatially homogeneous stable state E3(u3, v3) becomes 2 2 2 2 2 unstable to perturbation with wave numbers k where k 2 k1 ; k2 ; k1 and k2 being the finite boundary wave number which 2 can be found as roots of the equation C(k ) = 0, [17]. Therefore, 2

k1;2 ¼

pffiffiffiffiffiffi ðd11 C22 þ d22 C11  d12 C21  d21 C12 Þ  K1 ; where K1 ¼ ðd11 C22 þ d22 C11  d12 C21  d21 C12 Þ2  4ðd11 d22  d12 d21 ÞðC11 C22  C12 C21 Þ 2ðd11 d22  d12 d21 Þ ð4:6Þ

.

0.85

0.8

Predator (v)

0.75

0.7

0.65

0.6

0.55

0.5

0

0.2

0.4

0.6

0.8

1

Prey (u) Fig. 1. Phase portrait of global stability around the positive interior equilibrium point E3(u3, v3) corresponding to a = 0.59, b = 0.0327, c = 0.15, d = 0.05,  = 1.0.

0.6

0.59

Predator (v)

0.58

0.57

0.56

0.55

0.54

0.53

0.14

0.16

0.18 Prey (u)

0.2

0.22

Fig. 2. Hopf-bifurcation around the positive interior equilibrium point E3 corresponding to a = 0.57, b = 0.0327, c = 0.15, d = 0.05 at

 = 1.1005.

L.N. Guin et al. / Applied Mathematical Modelling 36 (2012) 1825–1841

1831

5. Conclusions and comments In the current investigation, we propose and analyse the spatial patterns through diffusion-driven instability in a predator–prey model. Here we consider a prey-dependent predator–prey interacting model with self as well as cross-diffusion and investigate the local asymptotic stability, global stability and bifurcation behaviour of the system. We analyse the system mathematically and describe its biological applications. The numerical simulations are carried out by the computer packages like MATLAB [18] and MAPLE [19] by making use of some experimental parameter values taken from Sherratt and Smith [20]. The first equilibrium point E0(0, 0) which represents the extinction of both the species is a saddle point if b < d. The stability of the equilibrium point E0(0, 0) has a considerable importance. If it would be stable, a non-zero population might be attracted towards it, and the system tends toward the extinction of both species. However, as the equilibrium point E0(0, 0) is

Fig. 3. Emergence of the Turing pattern corresponding to G = [(d11C22 + d22C11)  (d12C21 + d21C12)]2  4(d11d22  d12d21)(det J).

Fig. 4. Characterization of the dispersal relation for d22 = 4.0.

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Fig. 5. Spatial distribution of prey at different instants with d22 = 4.0 (assuming initial approximation as u0 = 0.007,

v0 = 0.003 and time interval dt = 0.06).

L.N. Guin et al. / Applied Mathematical Modelling 36 (2012) 1825–1841

1833

Fig. 6. Spatial distribution of prey at different instants with d22 = 12.0 (assuming initial approximation as u0 = 0.007, v0 = 0.003 and time interval dt = 0.06).

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Fig. 7. Spatial distribution of prey at different instants with d21 = 2.0 (assuming initial approximation as u0 = 0.007,

v0 = 0.003 and time interval dt = 0.06).

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a saddle point, and hence unstable, we find that the extinction of both the species is not possible in this model. Also the system enters into transcritical bifurcation at b = b[tc], where b[tc] = d, around the equilibrium point E0(0, 0).

At the axial equilibrium point E1 ð1; 0Þ the system is locally asymptotically stable iff b þ

c 1þa

 d < 0:

ð5:1Þ

It is globally asymptotically stable under certain conditions which are obtain through the construction of a local and global Lyapunov function using radial basis functions, [13,14]. Also the system enters into transcritical bifurcation at c = c[tc], where c[tc] = (d  b)(1 + a) around the equilibrium point E1(1, 0). Under the condition (5.1) predators intraspecies competition makes them tend toward extinction. At the equilibrium point E2 ð0; 1  bdÞ, the system (2.2a)–(2.2c) is locally asymptotically stable iff

 a d 857 the Turing pattern emerges and it can be shown that for d21 > 1.89 the Turing pattern also emerges. In order to estimate the effect of the parameter d22 (self diffusion coefficient of predator) the dispersion relation has been plotted corresponding to the system parameters a = 0.59, b = 0.0327, c = 0.15, d = 0.05,  = 1.0, d11 = 0.1, d12 = 0.01, d21 = 0.3, and d22 = 4.0 satisfying our assumption d11d22 > d12d21 which indicates that self diffusion is stronger than cross diffusion, i.e. the flow of the respective densities in the spatial domain depends strongly on their own density than on the others (cf. Fig. 4). The values of d12 and d21 imply that the prey species approaches towards the lower concentration of the predator species and the predator species tends to diffuse in the direction of lower concentration of the prey species. This type of incident occurs in nature where the prey approaches towards the lower concentration of the predator in search of new food and the predator prefers to avoid group defence by a huge number of prey and chooses to catch its prey from a smaller concentration group unable to sufficiently resist. Thus the effect of cross-diffusion in a predator–prey system plays significant role as pattern emerges. The Tables 1–6 represent contour pictures of spatial pattern through Turing instability to the present system (2.2a)–(2.2c) for different values of diffusion coefficients.

Fig. 9. Spatial distribution of prey at different instants with d12 = 0.013 (assuming initial approximation as u0 = 0.007, v0 = 0.003 and time interval dt = 0.06).

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Fig. 10. Spatial distribution of prey at different instants with different d11 (assuming initial approximation as u0 = 0.007, dt = 0.06).

v0 = 0.003

and time interval

Table 1 The table shows the values of diffusion coefficients, grid points (2D), number of iterations and the corresponding pictures of patterns. Values of diffusion coefficients

Grid points (2D)

No. of iterations

Contour pictures

d11 = 0.1, d11 = 0.1, d11 = 0.1, d11 = 0.1, d11 = 0.1, d11 = 0.1,

525  525 324  324 144  144 144  144 144  144 144  144

06 20 50 100 200 500

Fig. Fig. Fig. Fig. Fig. Fig.

d12 = 0.01, d12 = 0.01, d12 = 0.01, d12 = 0.01, d12 = 0.01, d12 = 0.01,

d21 = 0.3, d21 = 0.3, d21 = 0.3, d21 = 0.3, d21 = 0.3, d21 = 0.3,

d22 = 4.0 d22 = 4.0 d22 = 4.0 d22 = 4.0 d22 = 4.0 d22 = 4.0

5(a) 5(b) 5(c) 5(d) 5(e) 5(f)

Table 2 The table shows the values of diffusion coefficients, grid points (2D), number of iterations and the corresponding pictures of patterns. Values of diffusion coefficients

Grid points (2D)

No. of iterations

Contour pictures

d11 = 0.1, d11 = 0.1, d11 = 0.1, d11 = 0.1, d11 = 0.1, d11 = 0.1,

144  144 144  144 144  144 144  144 144  144 144  144

20 35 100 200 400 600

Fig. Fig. Fig. Fig. Fig. Fig.

d12 = 0.01, d12 = 0.01, d12 = 0.01, d12 = 0.01, d12 = 0.01, d12 = 0.01,

d21 = 0.3, d21 = 0.3, d21 = 0.3, d21 = 0.3, d21 = 0.3, d21 = 0.3,

d22 = 12.0 d22 = 12.0 d22 = 12.0 d22 = 12.0 d22 = 12.0 d22 = 12.0

6(a) 6(b) 6(c) 6(d) 6(e) 6(f)

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The results of Figs. 5–10 show the spatial pattern distributions of prey at two-dimensional (2D) space domain. The equations (2.2a)–(2.2c) describe the dynamics of a predator–prey community being assumed homogeneous. All the spatial patterns of these figures show the evolution of the prey species at different iterations, with very small random perturbation of the stationary solution u3 and v3 of the spatially homogeneous systems. The choice of the length of the domain may be different in different computer experiments. Here we have plotted the spatial pattern figures in normal finite domain i.e. ð0 0. Thus the equation (A.1) reduces to the following factorisation 3 X

Ai zi ¼ A3 ðz2 þ a1 z þ b1 Þðz þ v 1 Þ ¼ A3 ½z3 þ ða1 þ v 1 Þz2 þ ðb1 þ a1 v 1 Þz þ b1 v 1 ;

ðA:3Þ

i¼0

where ða1 þ v 1 Þ ¼ AA23 ; ðb1 þ a1 v 1 Þ ¼ AA13 ; b1 v 1 ¼ AA03 and from which one may easily get The other root of (A.3) is thus obtained as z3 = v1. By imposing bility of the interior equilibrium, that is,

A1 b1  < 0; a1 A3 a1 A0 < 0; ju1 j2 A3 A2 þ 2Reðu1 Þ < 0: A3

v 1 ¼ AA

2 3

þ 2Reðu1 Þ or

A1 a1 A3

 ba11 or

A0 . ju1 j2 A3

v1 < 0, we obtain z3 > 0 ensuring uniqueness and feasiðA:4aÞ ðA:4bÞ ðA:4cÞ

Also when the equation (A.1) has three real roots, we investigate the sufficient conditions for it to have exactly one real positive root. For the interior equilibrium point E3(u3, v3), let u3 be a real positive root of the cubic

a0 x3 þ 3a1 x2 þ 3a2 x þ a3 ¼ 0 ða0 – 0Þ:

ðA:5Þ

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Here a0 = b, 3a1 = 2ab  b, 3a2 = b  2ab + c  d + a2b, and a3 = ab  a2b  ad. Equation (A.5) is now reduced to

z3 þ 3Hz þ G ¼ 0;

ðA:6Þ

by the transformation z = a0x + a1. Equation. (A.5) has exactly one real positive root if G2 + 4H3 > 0, where G ¼ a20 a3  3a0 a1 a2 þ 2a31 ; H ¼ a0 a2  a21 and using Cardan’s method, we obtain the root as  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 ðGþ ðG2 þ4H3 ÞÞ . the three values of 2

r 1 rH a1 1

a0

, where r1 denotes one of

Appendix B Consider the differential equation (2.2a)–(2.2c) without the diffusive terms as

du ¼ f ðu; v Þ; dt dv ¼ gðu; v Þ; dt

ðB:1Þ ðB:2Þ

uv c uv  where f ðu; v Þ ¼ uð1  uÞ  uþ a and gðu; v Þ ¼ bv ð1  v Þ þ uþa  dv . Let x3 = (u3, v3) be an equilibrium of the system (B.1) and (B.2), such that all eigenvalues of J at x3 have negative real part. We calculate here the local Lyapunov function L3(u,v) for P = 2 (i.e. homogeneous quadratic polynomial) where P denotes the degree of the polynomial L3(u, v).

L3 ðu; v Þ ¼



u  u3

T

v  v3

 Q

u  u3

v  v3

;

where Q is the solution of

J T Q þ QJ ¼ I:

ðB:3Þ

In our case,

2 J¼4

1  2u3  ðuv 3þaaÞ2 3

acv 3

ðu3 þaÞ2

3

 5; Q ¼ k11 k21 bð1  2v 3 Þ þ uu33þca  d u3  ðu3 þaÞ

k12 k22



ðsayÞ and I ¼



1 0 0 1

:

One can obtain k11, k12 and k22 by solving the above Equation (B.3). Now L3(u,v) is positive definite if the leading minors of the square matrix Q are all positive that is if k11 > 0, k22 > 0 and k11k22  k12k21 > 0. Therefore

L3 ðu; v Þ ¼ k11 ðu  u3 Þ2  2k12 ðu  u3 Þðv  v 3 Þ þ k22 ðv  v 3 Þ2 ; for P = 2 [13,14] which has been depicted in Fig. 11 for a set of parameter values. Then it is possible to find a compact set K A(x3) with a positively invariant neighbourhood B1, where B1 itself is some neighbourhood of x3 such that x3 2 K. Moreover, L03 ðxÞ < 0 holds for all x 2 Kn{x3} and the local Lyapunov basin K = {x 2 B1jL3(x) 6 R} with R > 0. Therefore Lyapunov function L3(x) have negative orbital derivative for all x 2 A(x3)n{x3}. Here x = (u, v),x3 = (u3, v3) and A(x3) is the basin of attraction of an asymptotically stable equilibrium point x3 defined by A(x3) :¼ {x 2 RnjStx ? x3 as t ? 1} and St is the flow operator [14] of the system (B.1) and (B.2) at x3. Generally when the local Lyapunov function is a quadratic form then the sublevel set K is ellipse. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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