Chase-Repulsion analysis for (2+1)-Dimensional Lotka-Volterra System

Chase-Repulsion analysis for (2+1)-Dimensional Lotka-Volterra System

S. M. Mabrouk

Department of physics and Engineering Mathematics Faculty of Engineering, Zagazig University, Egypt.

AbstractHomogeneous balance method (HB) is applied to the (2+1)-dimensional Lotka-Volterra system to construct exact solutions. A homogeneous system of equations for the quasi- solution is solved. The travelling wave quasi-solution leads to the solitary wave solution of the system. According to the different system parameters, three ecosystems; predator-prey, symbiosis and competition are analyzed and plotted. The chase-repulsion relationship is cleared in the two opposite soliton waves for the predator-prey case. The inhibit and favor between species are obvious in competition and symbioses cases.

Keywords Lotka-Volterra equations; Homogeneous balance method; predator-prey dynamics; Mathematical ecology.

1. INTRODUCTION

   Investigation of the interaction between species plays an important role in mathematical ecology [1-8], technology evolution [9-11] and other applications [12, 13]. There are

   homogeneous balance method [44-47] . This work is motivated to solve and analyze the (2+1)-dimensions Lotka- Volterra system. The homogenous balance method is applied to find the exact soliton solution of this system. The paper is arranged as follows. In section II, the homogenous balance method is described. In section III, the method is applied to solve the Lotka-Volterra system in (2+1)- dimensions. Section IV, is devoted to analyze and discuss the results. The paper ends with conclusions in section V.

2. DESCRIPTION OF THE HOMOGENEOUS BALANCE METHOD

   The homogeneous balance method [44-47], is a systematic and effective for finding explicit solitary wave solutions. Consider a system of partial differential equations (PDE);

   1(, , , , , , . ) = 0

   six interaction modes between species in an ecosystem [14], namely, neutralism, competition, symbioses,

   {

   2(, , , , , , . ) = 0

   (1)

   commensalism, amensalism and predation (predator-prey). Predator-prey dynamics has been extensively studied by

   Lotka-Volterra models [15-18]. Many authors have focused on solving and analyzing the Lotka-Volterra models. Among them, Alam and Tunc [19], constructed many families of exact solutions of nonlinear Predator-prey equations using the exp (-()) expansion method. Kraenkel et al [20], Applied the expansion method to drive exact

   Where p1 and p2 are polynomials in , and their partial

   derivatives. A function = (, , ) is considered as a quasi-solution of the system (1) if there are functions f = f() and = (), of only one argument so that, a nominated linear combination of; 1, (), (), (), (), (), (), () . (2)

   and;1, (), (), (), (), (), (), () . (3)

   solutions to a predator-prey

   with Allee effect prey per capita

   Are solutions of the system (1). The HB method is recapped

   in the prosecuting steps;

   growth rate. They discussed the system for two different wave speeds and reported three different solutions for each wave velocity. In [21], the Khasminskiis theory of periodic solution, was applied to identify that the system fulfils a nontrivial positive T -periodic solution. Numerical simulations in different cases were analyzed [22-24]. The existence of traveling wave solutions for a diffusive predatorprey system were discussed by using the original Wazewskiis theory [25]. Different pattern formations for the Predator-prey system were discussed in [26-28].

   The plenteous number of mathematical methods of solutions of differential equations (DEs) [15, 29-43], authorized scientists to pointedly elucidate the different mathematical models. Some of these methods are; singular manifold method [29, 30], Hirotas bilinear method [31, 35], transformed rational function method [39, 41], exponential function method [36], Darboux transformation [32, 40], conservation laws and symmetry methods [42, 43], homotopy analysis transformation method [33, 34], direct

   Step 1: Choose the solution of (1) as a linear combination of

   (2) and (3), satisfying the balance between the highly nonlinear and the highest order derivative terms in the system (1).

   Step (2): Substitute the combination picked out in step 1 into the system (1). Collect all terms with the highest degree of (x, y, t) and set their coefficients to zero. Secure a system of ordinary differential equations (ODEs) in f() and (), solve this system to find f(), () and relations between their nonlinear derivatives.

   Step (3): Replace for nonlinear derivatives of () and f(), then collect all the terms with the same order of

   , , , , , , , , , , and set their coefficients to zero. Get a homogeneous system of ODEs in (x, y, t). According to the homogeneous property of this system of equations (x, y, t) can be predicted as an exponential function.

   Step (4): Substiute f(), () and (x, y, t) in the linear combination from step(1), then the solution of the system (1)

   algebraic method [37],

   expansion method [38] and

   is obtained.

3. SOLITARY WAVE SOLUTION OF (2+1)- DIMENSIONAL LOTKA-VOLTERRA SYSTEM

2 = , = 1

1 1

1 1

2

2 = , = 1

2 2 2

(15)

The (2+1) dimensional Lotka-Volterra system is represented as;

1 + 1 + (1 ) 1 = 0 (4)

= = 1 = 1

1 2

1 2

2 2

{ = 1 = 2

2 + 2 + 2 = 0 (5)

By using (12) and (15) the equations (10) and (11) are

simplified as;

Where (x, y, t) and (x, y, t) are the species densities, d1

(

+ 6

+

+ +

and d

are specific diffusion rate, r is the population intrinsic

1 2

1

1

1

2 ) 2 + (4

+ 4

+ 2

rate of growth, k is the carrying capacity, is the per capita

1

1

1

injury rate and s represent consumption interaction

2) + + 14 + 1 + 312 +

2 2 + 2 + 2 +

coefficient between species.

1

1

1

In this section the homogeneous balance method is 122 + 1 = 0

applied to find the exact solution of the Lotka-Volterra system (4)-(5). Choose the solution of this system as a linear combination in the form;

(16)

(21 + 62 + 62

)2 + (4 + 4 + 2

1 ()

2

2

2

(, , ) =

(6)

2) + 22 212 + 322 +

2

2

1

(, , ) = ()

(7)

2 + 22 + 222

2

+ + = 0

The balance between the highly nonlinear and the highest

order derivative terms in the system (4)-(5), results in 1 =

(17)

2 4

2

2 = 2. Then the linear combination (6) and (7) is written as;

setting the coefficents of , in (16) and the coefficients , in (17) equal to zero; yields a

(, , ) = 2 + (8)

system of partial differential equations for (x, y, t);

(, , ) = 2 +

(9)

Substitute (8) an (9) in 4);

(

+ 6

+ 1

+ ) 2 +

d to (

1 2

1

1

( 2 + (4) ) 4 + ( +

41 = 0 (18)

1

1

1

2 + (4

+ 2

+ 2

)

+ 3 2 +

(4)2

+ 6

2

1

1

1

1

1

1

2 2 + 2

+ 1 2 + 2 + = 0

1

+ 1

+ ) 2 + (4

+

1

(19)

1

1 2

1

1

41 + 21 2) + +

14 + 12 + 1 + 312 +

14 + 1 = 0 (20)

(21 + 62 2 )2 + 22 +

2 2 + 2

22

42 = 0 (21)

1

1

2 + (4

2

2

)

2 2

12 + 1 = 0

2

2

2 1

+ 2

+ 2 2

+ 3 2 = 0

(10)

Substitute (8) and (9) into (5);

2

(22)

2

2

1

(2 + 2(4))4 + (2(4)2 + 2 +

+ 24 2 = 0 (23)

2 + 62 + 2 )2 + (42 + 42 + 22 2 ) +

22 + 222 + 22 + 2 +

To solve the homogeneous system (18)-(23), assume that

(, , ) = 1 + +++ (24) Where , , are to be determined and is a constant.

22 + 222 +

24 + 2 = 0

(11)

Setting the coefficient of 4 = 0, after setting

= ,

Substitute (24) into the system (18)-(24), considering that

= . It is found that (, , ) satisfies this system of equations when, = 2d1, 1 = 2, =22d2 and = .

2

yielding a system of ordinary differential equations;

2 + 2 (4) = 0

Then the solution of the Lotka-Volterra system (4)-(5) is;

2

2

(, , ) = 1 secp (1 ( + + + )) (25)

{ 1 1

(12) 4 2

+ 2 (4) = 0

22 2 1

2 2 (, , ) = sech ( ( + + + )) (26)

The solutions of this ODE system are; 4 2

= ,

= 122

(13)

1 1 2

= ,

= 122 121

(14)

2 2 12

1

The relations between the nonlinear derivatives of () and

f() are summarized as;

1. RESULTS AND DISCUSSION

   This section is motivated to plot and discuss the solutions of the system (4)-(5). The dynamics of the species densities are varied according to the system parameters. It is clear that the relation between species is affected by the signs of the interaction coefficients (s). Table 1 illustrate three relations between species for different (s) signs.

   TABLE I. THE EFFECT OF THE INTERACTION COEFFICIENTS SIGNS IN THE RELATION BETWEEN SPECIES.

   Relation between species	Sign of 1	Sign of 2	Sign of interaction term in equation (4)	Sign of interaction term in equation (5)
   Prey-Predator	+	+		+
   Symbiosis		+	+	+
   Competition	+

   The prey-predator relation is presented in Fig. 1, for t = 0, t

   = 5 and t =10 with the parameters 1 = 2 = 50, d1 =10 , d2

   = 5, k = 10, r = 30, = 10, = 1, = 1, =1 and = 1. The densities evolve in two opposite solitary waves, which confirms the chase-repulsion relation between species. The increase in the prey density reveals a decrease in predator density and vice versa. The two dimensional plot for this case is shown in Fig.2, at y =2.

   1. (b) (c)

      Fig. 2: The 2D-prey- predator evolution at 1=2=50, d1=10, d2=5, k=10, r=30, =10, =1, =1, =1, =1 and y=2 for t =0, t =5 and t =10.

      The symbiosis relation between species appears when 1 is negative. Fig.3, represents the symbiosis relation between and for 1=100, 2=50, d1=10, d2=5, k =1, r =30, =10, =1, =1, =1, =1, t=2 and y=1. The solitary waves show that both species favor each other.

      The competition relation between and is illustrated in

      Fig.4, for 1 =100, 2 =50, d1 =100, d2 =50, k=1, r=30,

      =10, =1, =1, =1, =1, t=2 and y=1. The solution shows that Both species inhibit each other.

      1. (b) (c)

         (d) (e) (f)

         Fig. 1: (a, b and c) is the prey evolution and (d, e and f) is the predator evolution for 1 = 2 = 50, d1 =10 , d2 = 5, k = 10, r = 30, = 10, = 1, = -1, =-1 and = 1 at t = 0, t = 5 and t =10.

         .

         Fig. 3: The symbiosis relation between and for 1=100, 2=50, d1=10, d2=5, k =1, r =30, =10, =1, =-1, =-1, =1, t=2 and y=1,(a)

         represents , (b) represents and (c) is the 2D solitons.

         (c)

         Fig. 4: The competition relation between and for 1=-100, 2=-50, d1=100, d2=50, k =1, r =30, =10, =1, =-1, =-1, =1, t=2 and

         y=1,(a) represents , (b) represents and (c) is the 2D solitons.

2. CONCLUSIONS

Homogeneous balance method is effective in detecting the solitary wave solution of the Lotka-Volterra system. Three interaction relations are discussed and plotted. The chase- repulsion relation between species is conspicuous in the two opposite solitary waves of the prey-predator case. The dynamics of the system is discussed for the different parameter values. The symbiosis and competition relations manifested at different signs of the interaction coefficients

1. M.N. Alam, C. Tunc, An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator-prey system, Alexandria Engineering Journal, 55 (2016) 1855-1865.

2. R.A. Kraenkel, K. Manikandan, M. Senthilvelan, On certain new exact solutions of a diffusive predator-prey system, Communications in Nonlinear Science and Numerical Simulation, 18 (2013) 1269-1274.

3. Q. Liu, D. Jiang, Periodic Solution and Stationary Distribution of Stochastic PredatorPrey Models with Higher-Order Perturbation, Journal of Nonlinear Science, 28 (2018) 423-442.

4. J. Wang, J. Shi, J. Wei, Dynamics and pattern formation in a

   1 and 2.

   REFERENCES

   diffusive predatorprey system with strong Allee effect in prey,

   Journal of Differential Equations, 251 (2011) 1276-1304.

5. M. Hafiz Mohd, Y. Abu Hasan, Modelling the spatiotemporal dynamics of diffusive prey-predator interactions: Pattern formation

   1. C. Lu, X. Ding, Periodic solutions and stationary distribution for a stochastic predator-prey system with impulsive perturbations, Applied Mathematics and Computation, 350 (2019) 313-322.

   2. Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Dynamics of a Stochastic PredatorPrey Model with Stage Structure for Predator and Holling Type II Functional Response, Journal of onlinear Science, 28 (2018) 1151-1187.

   3. H. Li, H. Xiao, Traveling wave solutions for diffusive predator- prey type systems with nonlinear density dependence, Computers & mathematics with applications, 74 (2017) 2221-2230.

   4. S.H. Piltz, F. Veerman, P.K. Maini, M.A. Porter, A Predator-2 Prey Fast-Slow Dynamical System for Rapid Predator Evolution, SIAM Journal on Applied Dynamical Systems, 16 (2017) 54-90.

   5. K.M. Owolabi, K.C. Patidar, Numerical simulations of multicomponent ecological models with adaptive methods, Theoretical biology and medical modeling, 13 (2016) 1-25.

   6. D.H. Nguyen, G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, Journal of differential equations, 262 (2017) 1192-1225.

   7. L.N. Guin, H. Baek, Comparative analysis between prey- dependent and ratio-dependent predatorprey systems relating to patterning phenomenon, Mathematics and Computers in Simulation, 146 (2018) 100-117.

   8. F. Capone, M.F. Carfora, R. De Luca, I. Torcicollo, On the dynamics of an intraguild predatorprey model, Mathematics and Computers in Simulation, 149 (2018) 17-31.

   9. G. Zhang, D.A. McAdams, V. Shankar, M.M. Darani, Technology Evolution Prediction Using LotkaVolterra Equations, Journal of Mechanical Design, 140 (2018) 061101.

   10. G. Zhang, D.A. McAdams, V. Shankar, M.M. Darani, Modeling the evolution of system technology performance when component and system technology performances interact: Commensalism and amensalism, Technological Forecasting and Social Change, 125 (2017) 116-124.

   11. G. Zhang, D. Allaire, D.A. McAdams, V. Shankar, System evolution prediction and manipulation using a LotkaVolterra ecosystem model, Design Studies, 60 (2019) 103-138.

   12. X. Dong, R. Ma, Analysis on the symbiosis stability of agricultural equipment manufacturing value network based on Lotka-Volterra, International Journal of Systems Assurance Engineering and Management, 8 (2017) 499-504.

   13. A. Marasco, A. Picucci, A. Romano, Market share dynamics using Lotka-Volterra models, Technological forecasting and social change, 105 (2016) 49-62.

   14. E.P. Odum, G.W. Barrett, Fundamentals of ecology, Cengage Learning, Australia, 2009.

   15. . Yüzba, M. Karaçayr, A numerical method for solutions of LotkaVolterra predatorprey model with time-delay, International Journal of Biomathematics, 11 (2018) 1850028.

   16. W. Ni, J. Shi, M. Wang, Global stability and pattern formation in a nonlocal diffusive LotkaVolterra competition model, Journal of Differential Equations, 264 (2018) 6891-6932.

   17. T. Dannemann, D. Boyer, O. Miramontes, Lévy flight movements prevent extinctions and maximize population abundances in fragile LotkaVolterra systems, Proceedings of the National Academy of Sciences, 115 (2018) 3794-3799.

   18. S. Vaidyanathan, Anti-synchronization of the generalized lotka- volterra three-species biological systems via adaptive control, International Journal of PharmTech Research, 8 (2015) 141-156.

and ecological implications, ScienceAsia, 39S (2013) 31-36.

1. M. Sambath, K. Balachandran, Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge, Journal of Applied Analysis and Computation, 3 (2013) 71-80.

2. X. Lin, P. Weng, C. Wu, Traveling Wave Solutions for a Predator- Prey System With Sigmoidal Response Function, Journal of Dynamics and Differential Equations, 23 (2011) 903-921.

3. W. Wang, Y. Lin, L. Zhang, F. Rao, Y. Tan, Complex patterns in a predator-prey model with self and cross-diffusion, Communications in Nonlinear Science and Numerical Simulation, 16 (2011) 2006-2015.

4. X. Lian, Y. Yue, H. Wang, Pattern Formation in a Cross-Diffusive Ratio-Dependent Predator-Prey Model, Discrete Dynamics in Nature and Society, 2012 (2012) 1-13.

5. X. Lian, S. Yan, H. Wang, Pattern formation in predator-prey model with delay and cross diffusion, in: Abstract and Applied Analysis, Hindawi, 2013, pp. 10 pages.

6. R. Saleh, M. Kassem, S. Mabrouk, Exact solutions of Calgero- Bogoyavlenskii-Schiff equation using the singular manifold method after Lie reductions, Mathematical Methods in the Applied Sciences, 40 (2017) 5851-5862.

7. S.M. Mabrouk, A.S. Rashed, Analysis of (3+ 1)-dimensional BoitiLeonMannaPempinelli equation via Lax pair investigation and group transformation method, Computers & Mathematics with Applications, 74 (2017) 2546-2556.

8. Y. Cao, B.A. Malomed, J. He, Two (2+ 1)-dimensional integrable nonlocal nonlinear Schrödinger equations: Breather, rational and semi-rational solutions, Chaos, Solitons & Fractals, 114 (2018) 99- 107.

9. R. Guo, H.Q. Hao, L.L. Zhang, Dynamic behaviors of the breather solutions for the AB system in fluid mechanics, Nonlinear Dynamics, 74 (2013) 701-709.

10. V.F. Morales Delgado, J.F. Gómez Aguilar, M.A. Taneco Hernández, D. Baleanu, Modeling the fractional non-linear Schrödinger equation via Liouville-Caputo fractional derivative, Optik, 162 (2018) 1-7.

11. V.F. Morales Delgado, J.F. Gómez Aguilar, D. Baleanu, A new approach to exact optical soliton solutions for the nonlinear Schrödinger equation, The European Physical Journal Plus, 133 (2018) 189.

12. L. Li, C. Duan, F. Yu, An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation, Physics Letters A, (2019).

13. S. Arshed, A. Biswas, F.B. Majid, Q. Zhou, S.P. Moshokoa, M. Belic, Optical solitons in birefringent fibers for Lakshmanan PorsezianDaniel model using exp ( ())-expansion method, Optik, 170 (2018) 555-560.

14. D. Lu, A.R. Seadawy, M. Arshad, Brightdark solitary wave and elliptic function solutions of unstable nonlinear Schrödinger equation and their applications, Optical and Quantum Electronics, 50 (2018) 23.

15. F. Batool, G. Akram, New solitary wave solutions of the time- fractional Cahn-Allen equation via the improved G'/G-expansion method, The European Physical Journal Plus, 133 (2018) 171-182.

16. R. Asokan, D. Vinodh, Soliton and Exact Solutions for the KdV BBM Type Equations by tanhcoth and Transformed Rational Function Methods, International Journal of Applied and Computational Mathematics, 4 (2018) 100-120.

17. N. Liu, X.Y. Wen, Y. Liu, Fission and fusion interaction phenomena of the discrete kink multi-soliton solutions for the ChenLeeLiu lattice equation, Modern Physics Letters B, 32 (2018) 1850211.

18. W.X. Ma, J.H. Lee, A transformed rational function method and exact solutions to the 3+ 1 dimensional JimboMiwa equation, Chaos, Solitons & Fractals, 42 (2009) 1356-1363.

19. A. Aliyu, M. Inc, A. Yusuf, D. Baleanu, Symmetry analysis, explicit solutions, and conservation laws of a sixth-order nonlinear ramani equation, Symmetry, 10 (2018) 341.

20. M. Inc, M.S. Hashemi, A. Aliyu, Exact Solutions and Conservation Laws of the Bogoyavlenskii Equation, Acta Physica Polonica, A., 133 (2018).

21. J. Ji, J. Wu, J. Zhang, Homogeneous balance method for an inhomogeneous KdV equation: Backlund transformation and Lax pair, International Journal of Nonlinear Science, 9 (2010) 69-71.

22. D. Lu, B. Hong, L. Tian, Backlund transformation and n-soliton- like solutions to the combined KdV-Burgers equation with variable coefficients, International Journal of Nonlinear Science, 2 (2006) 3-10.

23. M. Wang, Y. Wang, Y. Zhou, An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications, Physics Letters A, 303 (2002) 45-51.

24. B. Li, Y. Chen, H. Zhang, Auto-Backlund Transformations And Exact Solutions For The Generalized Two-Dimensional Korteweg- De Vries-Burgers-Type Equations And Burgers-Type Equations, Zeitschrift für Naturforschung A, 58 (2003) 464-42.