Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion

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1 Applied Mathematis, 013, 4, Published Online August 013 ( Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion Muhammad Shakeel Department of Mathematis, Air University, Islamabad, Pakistan Reeived May 11, 013; revised June 11, 013; aepted June 19, 013 Copyright 013 Muhammad Shakeel This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited ABSTRACT In this paper we study one-dimensional Fisher-Kolmogorov equation with density dependent non-linear diffusion We hoose the diffusion as a funtion of ell density suh that it is high in highly ell populated areas and it is small in the regions of fewer ells The Fisher equation with non-linear diffusion is known as modified Fisher equation We study the travelling wave solution of modified Fisher equation and find the approximation of minimum wave speed analytially, by using the eigenvalues of the stationary states, and numerially by using COMSOL (a ommerial finite element solver) The results reveal that the minimum wave speed depends on the parameter values involved in the model We observe that when diffusion is moderately non-linear, the eigenvalue method orretly predits the minimum wave speed in our numerial alulations, but when diffusion is strongly non-linear the eigenvalues method gives the wrong answer Keywords: Fisher-Kolmogorov Equation; Non-Linear Diffusion; Travelling Wave; Wave Speed; Pulled Front; Pushed Front 1 Introdution For a long time diffusion has been used as a well known model for spatial spread in many biologial systems Okubo [1] and Murray [] used diffusion in invasion and pattern formation and Skellam [3] studied diffusion in the field of eology For motile ell populations diffusion has been used in different situations Sherrat and Murray [4] used diffusion to model wound healing and Chaplain and Stuart [5] used it to model the apillary growth network To study the spatial movement of ell populations, linear diffusion is one of the well known model However, for losely paked ells suh as epithelia, linear diffusion model is unsuitable to examine many biologial systems inluding the movement of ell populations [6] For losely paked ells, a single ell population an be modeled by a reation diffusion equation [4,7], but for dissimilar ell populations, a diffusion term would mean that different ell populations an mix together ompletely and movement of one ells type an be influened by the ells of other type But atually the situation is totally opposite, different ell populations annot move through one another: instead the ell will stop moving when it suddenly omes aross another ell This phenomenon is known as ontat inhibition of migration This proess has been well doumented in many types of ells [8] In all biologial systems the exhange of information at both inter- and intra-ellular level is almost ontinuous In order to get sequential development and generation of the required pattern suh ommuniation is neessary eg for development and growth of an embryo Propagating waveforms are one of the ways of onveying suh biologial information between the ells Let us onsider a simple one-dimensional diffusion equation N N D, (1) t x where N is hemial (ell or nutrient) onentration and D is diffusion oeffiient The time to exhange information in the form of hanged onentration is OL D, where L is the length of domain We an get this order by dimensional arguments of Equation (1) At the early stage of growth of an organism the diffusion oeffiient an be very small: values of order 10 9 to m se 1 If diffusion is the main proess to onvey the biologial information then to over marosopi distanes of several millimeters requires a very long time When the diffusion oeffiient is O (10 9 to m se 1 ) and L is order of 1 mm then the time Copyright 013 SiRes

2 M SHAKEEL 149 required to onvey the information is O (10 7 to 10 9 se), whih is very large in the early stages of growth of an organism This means that simple diffusion is unlikely to be the main means of exhanging the information during embryogenesis Karevia [9] and Tilman and Karevia [10] estimated the diffusion oeffiient for inset dispersal in interating population About seventy years ago Fisher [11] and Kolmogorov et al [1] introdued a lassial model to desribe the propagation of an advantageous gene in a one-dimensional habitat The equation desribing the phenomenon is a one-dimensional non-linear reation-diffusion equation, N N D N 1N, () t x where N is hemial onentration, D is the diffusion oeffiient and the positive onstant represents the growth rate of the hemial reation Sine then a great deal of work has been arried out to extend their model to take into aount the other biologial, hemial and physial fators The Equation () is also used in flame propagation [13], nulear reator theory [14], autoatalyti hemial reations [15,16], logisti growth models [17] and neurophysiology [18] One of the extensions of the Fisher and Kolmogorov model is to introdue a non-linear diffusion oeffiient, whih an be taken as a non-linear Fik s law The nonlinearity an arise in terms of spae, time or density-dependent diffusion oeffiient In the Fisher and Kolmogorov model the reation kinetis are oupled to diffusion whih gives travelling waves of hemial onentration and it an affet biologial hange very muh faster as ompared to the proesses governed by simple diffusion without the kineti term In many biologial populations density-dependent dispersal has been observed eg Carl [19] observed that ground squirrels move from highly populated area to sparsely populated areas, Myers and Krebs [0] studied the population density yles in small rodents Several mathematial models have been developed to desribe the density-dependent dispersal systems Gurney and Nisbet [1,] developed a first density-dependent diffusion model in eologial ontext by using a random walk approah and Montroll and West [3] developed a one-dimensional model for a single speies by using the same approah But there is not muh work on the Fisher equation with non-linear diffusion Sánhez and Maini [4] studied a travelling wave solution in a degenerate Fisher equation with non-linear diffusion Although Equation () is known as the Fisher-Kolmogorov equation, the disovery, investigation and analysis of travelling waves in hemial reations was first reported by Luther [5] He found that the wave speed is a simple onsequene of the differential equations He obtained the wave speed in terms of parameters assoiated with the reations he was studying The analytial form is the same as that found by Kolmogorov [1] and Fisher [11] Seeking the travelling wave solution of Fisher equation has been a hallenging and diffiult task Several authors used different methods to find the travelling wave solution of Fisher equation Ablowitz and Zeppetella [6] obtained the first expliit form of travelling wave solution of Fisher equation using Painlevé analysis The authors found that kink wave propagates from left to right with a speed v 5 6 Wazwaz [7] used tanh method to find the travelling wave solution of non-linear partial differential equation The author also extended this study to the equations whih do not have tanh polynomials He also demonstrated the effiieny of the method by applying it to variety of equations eg Fisher equation Mansour [8] found a travelling wave solution of lass of degenerate reation diffusion equations in whih non-linearity is assumed to be singular at zero He employed two methods to find the wave profile and speed of the front These method inlude travelling wave equation and initial boundary value problem with an adaptive travelling wave ondition for partial differential equation Tan et al [9] used an analytial method namely Homotopy analysis method to solve the Fisher equation and they desribed a family of travelling wave solutions Jiaqi et al [30] found a travelling wave solution of non-linear reation diffusion equation by using the homotopi method and theory of travelling wave transform Feng et al [31] found the travelling wave solution of reation diffusion equation by two methods Firstly they applied Divisor theorem for two variables in omplex domain to find a quasi-polynomial first integral of an expliit form to an equivalent autonomous system Then through this first integral, they redued the reation-diffusion equation to a first-order integrable ordinary differential equation, and found a lass of traveling wave solutions They ompared their results with the existing results and found some errors in analyti results in literature They larified and introdued the refined results Hou et al [3] studied the travelling wave solution of reation diffusion equation with double degenerate nonlinearities They investigated existene, uniqueness and stability of wave solution In this paper we study the growth of ells in one-dimensional saffold in the presene of onstant nutrients We model a system in whih ell growth and diffusion takes plae simultaneously but the ell diffusion depends on ell density It inreases with inrease in ell density The equation governing this system is a non-linear Fisher equation in whih the diffusion oeffiient depends on ell density The non-linearity arises in the diffusion oeffiient We represent the diffusion oeffiient as an Copyright 013 SiRes

3 150 M SHAKEEL exponential funtion of ell density The form of nonlinear diffusion aptures the feature that it is very small for small ell density and inreases with the inrease in ell density and it is imum when ells stop growing Results are presented for different values of the parameters The Fisher equation has suh wide appliability in itself but also it is the prototype equation whih admits travelling wavefront solutions We study the travelling wave solution of this equation and find the approximation for the minimum wave speed We find that the minimum wave speed depends on the model parameters For moderately non-linear systems the analytial method orretly predits the wave speed in our numerial alulations However for strongly non-linear diffusion the numerial wave speed beomes larger than the analyti minimum speed The paper is organized as follows in Setions and 3 we present the dimensional and nondimensional model equations respetively The parameter values and travelling wave solution are desribed in Setions 4 and 5, respetively We present the numerial solution in Setion 6 and alulate the numerial minimum wave speed in Setion 7 Mathematial Model In this Setion we model ell growth in a bioreator subjet to uniform nutrient onentration Consider the ells are seeded onto a porous saffold with interonneted pores, whih is plaed in the bioreator The saffold extends from L x L, where x denotes the spatial oordinate In this model we represent the ell density by N, initial ell density by N, imum N init arrying apaity of the system by and diffusion oeffiient by D We assume that the environment is inhomogeneous ie the ell density N and diffusion oeffiient D depends on spatial oordinate x and time t, ie N N x, t and D D N We know that the ells require some nutrients suh as oxygen and gluose et to live and perform speifi funtions Suppose that the onentration of suh nutrients remains uniform everywhere in the entire domain at all times We want to model a system in whih the hange in ell density is due to ell proliferation and ells disperse in the entire domain due to diffusion We assume that when the ell density is small diffusion is also very small and N when the ell density reahes its imum arrying apaity N the ell proliferation stops and ells spread in the entire domain via diffusion For that we onsider a logisti growth model in whih the ell population spreads via diffusion That means we have a oupled system of reation kinetis and diffusion Our modelling of ell proliferation [33,34] has led us to the following Fisher equation with non-linear diffusion N N N D N N 1, (3) t x x N where is the imum ell growth rate and D N is the non-linear diffusion We observe that ells grow in numbers due to seond term on the right hand side of Equation (3) and they disperse in the entire domain due to first term on the right hand side of Equation (3) To model ell proliferation we need to hoose a form of D N so that, in regions of low ell density, ells grow by inreasing their density with little or no spatial motion, whereas in regions of high ell density the newly formed ells diffuse rapidly towards regions of lower ell density Thus to model this behaviour on a ontinuum level we hoose the following form for the non-linear diffusion n D N D exp N N, (4) where parameter D n is the ell diffusion when ell density N reahes its imum arrying apaity N, so we an all this the imum value of ell diffusion and the parameter ontrols how fast ells are spreading in the domain The parameters D n and have dimensions m /se and m 3 /ell respetively The exponential funtion in Equation (4) ensures that ell diffusion will remain positive The non-linear ell diffusion D N also has imum value D n when 0 So we an say that non-linear ell diffusion is imum either when ell density N reahes its imum arrying apaity N or the value of parameter is zero The non-linear diffusion beomes linear when 0 It is also lear from the Equation (4) that when there are no ells then in that ase ell diffusion is D n exp N, whih is the minimum value of diffusion So ell diffusion varies in the range, D exp N D N D (5) n Initially when the ell density N is small, the diffusion oeffiient D N is also small and D N inreases with inreasing ell density N but it always remains positive The initial ell density N x,0 is given by N x,0 N x (6) init 3 Nondimensionalization For onveniene and to redue the parameters in the equation we resale all the variables to analyze the nondimensional form of the equation We nondimensionalize all lengths by L whih we take as a typial length sale for the problem and all ell densities by the imum arrying apaity N n Copyright 013 SiRes

4 M SHAKEEL x L x,, (7) L N N N, N N N (8) init We nondimensionalize time by the speed of the growth front v (whih is given by Equation (30)), L t t (9) v In dimensionless form Equation (3) an be written as N N DN N1 N, 1 x t x x 1, (10) where DNexp 1 and N The parameters and are dimensionless parameters in the model whih are given by Dn L,and (11) Lv v The parameter is the ratio of ell growth rate to speed of growth front and parameter is the ratio of imum value of non-linear diffusion to the speed of growth front The parameter ontrols the diffusion and the parameter ontrols the growth term The initial ondition (6) in dimensionless form beomes N x,0 N x (1) 4 Parameter Values The modified Fisher Equation (10) inludes a number of parameters Some parameters depend on the ell type and some parameters depend on saffold geometry Table 1 shows the values of the parameters used in the simulation We assume that the length L of the saffold is 00 m Some quantities suh as ell growth rate depend on the ell type ultured in the bioreator The above proposed model is a generi model and an be applied to any ell type To ompare the model with the experimental data the ells used in the simulations are Murine immortalized rat ell CC 1 The imum ell growth rate for CC 1 ells is [35] Sine ell growth is a slow proess we an hoose that speed of growth front init init v is very small eg 10 or 1 or 01 mm day The value of dimensionless parameter an be obtained by using the values of dimensional parameters, L and v The values of parameters and are not available in the literature To estimate the value of these parameters we use the expression for dimensionless wave speed ie v exp (whih we will alulate later in Setion 5) We hoose that the theoretial wave speed v 1 In the expression for v there are two unknowns and In order to find the value of and we fix one of the parameters or in the expression for v If we fix the parameter then using the expression for v we an find the value of parameter We observe that the value of dimensionless parameter depends on the value of parameter If the value of parameter is high then value of is also high This means that ells will diffuse more quikly for high values of parameter Table 1 shows the values of the dimensional parameters used in the model We know that the initial ell density,0 N init x We an use any form of N x init 5 Travelling Wave Solution N x We are interested in travelling wave solutions beause Equation (10) has two steady states N 0 and N 1, whih are, respetively, unstable and stable This suggests that we should look for a travelling wave solution of Equation (10) for whih 0 N 1; sine negative ell density has no physial meaning If a travelling wave solution exists it an be written in the form N x, t, x vt, (13) where v is the wave speed Sine Equation (10) is invariant if x x, v may be negative or positive To be speifi we assume that v 0 We have N d, N d,and N d v t d x d x d (14) Substituting the travelling wave solution (13) into Equation (10) and using relations from Equation (14) we get a seond order ordinary differential equation, Table 1 Model parameters used in this work Parameter Desription Value Unit L Saffold length 001 m N Maximum arrying apaity Maximum ell growth rate ells/m /se v Speed of growth front mm/day Copyright 013 SiRes

5 15 M SHAKEEL where dd D v 10, (15) d D exp 1, (16) where primes denote the differentiation with respet to A typial wave front solution is a solution suh that approahes one steady state as and approahes the other steady state as Therefore the boundary onditions for travelling wave solution are lim 0, lim 1 (17) 51 Phase Plane Analysis From Equation (15) we observe that we have to determine the values of the wave speed v suh that a nonnegative solution of Equation (15) exists We use phase plane analysis to haraterize solutions of Equation (15) in the, phase plane where,, (18a) dd d D v 1 (18b) The system of ordinary differential Equations (18) is nearly singular at 0, sine D 0 0 for high values of parameter To remove the near singularity we introdue a new parameter [4], in suh a way that d 1 ds d D (19) 0 D s Exept at Thus we have and we obtain 0, where d d is not defined, d 0 d, (0) d d d d and d d d d and (1) Substituting into the system of Equations (18) we get D f,, (a) 1 dd 1 d g,, (b) v where dot denotes the derivative with respet to The phase trajetories of () are solutions of dd v 1 d d (3) d D The fixed points s, s are the points where f, 0 g,, these are steady states So in this ase fixed points are 0,0 and 1,0 The loal behaviour of the trajetories of system () an be obtained by analyzing the linear approximation of system () around eah fixed point 5 Stability of Fixed Points The system of Equations () has two fixed points, but whih of these fixed points are stable? The loal stability of a fixed point s, s is determined by linearization of the dynamis at the intersetion So with the linear approximations the system of Equation () beomes f f ab, (4a) g g d, (4b) whih an be written in the matrix form as A, (5) where f f a b, A, (6) d g g Let 1 and be eigenvalues of A then we have where det A I 0, a d A det 0 v D D 1, (7) D v D 4de A t Eigenvalues for the fixed points (0, 0) and (1, 0) are 1 v v 4D 0 0,0 :, (8a) 1 v v 4D 1 1,0 : (8b) D 0 exp and D 1 1 Us- We know that Copyright 013 SiRes

6 M SHAKEEL 153 ing values of D 0 and D 1 in Equation (8) we get 1 v v 4 0,0 : exp, 1 v v 4 1,0 : (9a) (9b) It is lear from the Equation (9a) that fixed point 0,0 is a stable node if v 4 exp vv exp, with the ase when v v giving a degenerate node The fixed point (0, 0) is a stable spiral if v 4 exp or vv exp ; ie osillates in the viinity of origin When v exp then it is not physially realisti beause Nx, t annot be negative The fixed point 1,0 is a saddle point The solution of our modified Fisher equation evolve to a travelling wave if the fixed point 0,0 is a stable node and minimum wave speed of wave front is v exp The ondition that 0,0 is a stable node is a neessary ondition for travelling wave propagation but not suffiient Sometimes it gives wrong answer as we shall in see Setion 7 If the propagation speed of the front is determined by the leading edge of the population distribution the non-linear fronts whose asymptoti propagation speed equals v is alled a pulled front In this ase the eigenvalues give the right answer beause the wave speed is determined by what happens at the front edge where 0 As above v exp is determined by the eigenvalues thus we all v speed of pulled front On the other hand if the speed of front is determined by the whole wave-front but not the behaviour of the leading edge the front is alled the Pushed front [36,37] In this ase simply omputing the eigenvalues gives the wrong answer for the minimum wave speed and the minimum wave speed is vmin v Thus the wave speed for a pulled front is given by v exp The wave speed v depends on the parameters, and, is diretly proportional to the square root of the produt of and In terms of the original dimensional Equation (3) the wave speed v is n v D exp N (30) We observe from the system of equations (8) that for a general funtion D the wave speed v is determined by D 0 So in general the wave speed of the Pulled front is v D0 Figure 1 shows the phase plane sketh of the trajetories of Equation () when v v We see that when Figure 1 Phase plane trajetories of Equation () Here parameter values are χ = , δ = γ = and v = 15 > v For explanation of parameter values see Setion 4 v v the fixed point 0,0 is a stable node beause all the trajetories from 1,0 to 0,0 have the same limiting diretion towards 0, 0 and the fixed point point 1,0 is a saddle point beause there are two inoming trajetories and two out going trajetories and all the other trajetories in the neighborhood of the ritial point 1,0 bypass 1,0 Similarly Figure shows the phase plane sketh of the trajetories of Equation () when v v We observe that when v v the fixed point 0,0 is a stable spiral beause all the trajetories from 1,0 to 0,0 spiral around the point 0, 0 Figures 3 and 4 show the phase plane sketh of trajetories of Equation () for various wave speeds v v and v v respetively We observe from the Figure 3 that when v v all the trajetories in the phase plane, from 1,0 to 0,0 remain entirely in the quadrant where 0 and 0, with 0 1 for all wave speeds v v Similarly from the Figure 4 we see that for all wave speeds v < v the phase trajetories from 1,0 to 0,0 spiral around the fixed point 0,0 In this ase osillates in the viinity of the origin giving negative whih is unphysial Copyright 013 SiRes

7 154 M SHAKEEL Figure Phase plane trajetories of Equation () Here parameter values are χ = , δ = 00139, γ = and v = 05 < v For explanation of parameter values see Setion 4 Figure 3 Phase plane trajetories of Equation () for different values of v v The other parameter values are same as in Figure 1 Eah urve (starting from bottom) represents the trajetory for various values of speed v v, ie v = 1, v = 15, v =, v = 5 and v = 3 respetively Figure 4 Phase plane trajetories of Equation (51) for different values of v < v The other parameter values are same as in Figure 1 Eah urve (starting from bottom) represents the trajetory for various values of speed v < v, ie v = 05, v = 06, v = 07 and v = 08 respetively 53 Seletion of Initial Condition A very important question at this stage is what kind of initial ondition N x,0 will evolve into the travelling wave solution and if the travelling wave solution exists what is its wave speed v? Fisher [11] found that Equation (10) has an infinite number of travelling wave solutions for whih 0 N x,0 1 for all wave speeds v v Kolmogorov [1] proved that Equation (10) has a travelling wavefront solution and the wave speed is v v, if N x,0 has ompat support A funtion N x,0 is said to have a ompat support if where N init x N x,0 N x 0, (31) init if 1 x F x x x 0 if x x x 1 where x1 x N x,0 Ninit x is ontinuous in x1, x If the initial ondition is other than (36) then solution depends on the behaviour of Nx,0 D then Equation (15) redues to If 1 and v 1 0 (3) A travelling wave solution of Equation (3) in expliit form for 1 was found by Ablowitz and Zep- Copyright 013 SiRes

8 M SHAKEEL 155 petella [6] for the speial wave speed v , 1 1 exp 6 (33) But if D is not a onstant then it is not possible to find the exat solution of Equation (15) Solution of suh non-linear problem an be approximated by perturbation theory or numerial investigation 6 Numerial Solution Numerially we solve the modified Fisher Equation (10) by using the ommerial finite element solver COMSOL We subdivide the domain 1 x 1 into a suitable number of mesh elements (intervals) of length x The end points of eah interval are alled node points and the elements do not have to have the same length But in this ase the length x of eah element is same To obtain meaningful results are is required in the hoie of a suitable number of mesh elements, finite element approximation and model parameters Convergene is ahieved by suessively refining the mesh elements The refined mesh ontains 30,71 mesh verties and 30,70 mesh elements The dependent variable N is approximated by a quadrati shape funtion and solved for 61,441 degrees of freedom We assume that at time t 0 the ell density is N and after the time init t tnew the ell density is Nnew We start with initial ell density N init and after eah time t new we replae Ninit by Nnew and solve the Equation (10) again for updated ell density The time from t 0 to t tnew is subdivided as t 0: t: t ne w, where t is the time step size from t 0 to t t and new t new is the time when we update the ell density The ell density N at eah mesh point x is obtained for different times To estimate the wave speed v numerially we look for the point x after eah time t where the ell density is half of its imum value ie N 1 imum ell densi ty When the ell density is half of its imum value at time t t, then 1 x x1 and at time t t, x x So we an estimate the total distane x x x1 traveled by the wave in the time interval t t t 1 Hene total distane x wave speed (34) total time t Results are plotted for different values of the dimensionless parameters and in Setions 61 to 64 In the next Setion we onsider various ases in whih we fix the value of dimensionless parameter χ and vary the values of dimensionless parameter and parameter in suh a way that the theoretial wave speed v N x N H r x, remains 1 Let us assume that init 0 where N 0 and r are onstants and H is the Heaviside step funtion 61 Case I: χ = 1317 In this ase we fix the value of dimensionless parameter 1317 and find the values of parameter and suh that theoretial speed of growth front v is 1 Table shows the values of dimensionless parameter for orresponding values of parameter Figures 5 and 6 show the numerial solution of the modified Fisher Equation (10) for the parameter values given in the Table We observe from Figures 5 and 6 that the solution does not evolve into a travelling wave solution before it reahes the domain edges The ell density N drops down beause the rate of diffusion is larger than the growth rate, whih means that the dimensionless parameter is larger than the dimensionless parameter Diffusion dominates in this simulation so we need to look at the ase where diffusion is smaller and ell growth is larger, to find a travelling wave Numerial results are plotted for 1 and in Figures 5 and 6 In a larger domain and with a longer time of solution the system will eventually evolve into a travelling wave but we are interested in a finite domain 6 Case II: χ = We onsider the ase when the dimensionless parameter Table Values of and for Figure 5 Numerial results of profile of ell density N at different times when γ = 1, δ = 05141, χ = 1317 Initial ell density is Ninit x N0Hr x, where N 0 05, and r = 01 The time step size Δt = 0001 and ell update time t new = 001 The figure shows the ell distribution after eah time t new and final time is t = 03 Copyright 013 SiRes

9 156 M SHAKEEL Table 3 Values of and for Figure 6 Numerial results of profile of ell density N at different times when γ =, δ = 13976, χ = 1317 N init, Δt and t new are same as in Figure and find the value of parameter and suh that theoretial speed of growth front v is 1 Table 3 shows the values of dimensionless parameter for the orresponding values of parameter Numerial results of modified Fisher Equation (10) are plotted in Figures 7 and 8 for parameter values given in Table 3 It is lear from the Figures 7 and 8 that in this ase the solution evolves into travelling wave fronts So in this ase the growth term is somewhat larger than the diffusion term But the diffusion term is not too small beause the ells also diffuse very quikly This feature is evident from the Figures 7 and 8 beause at the edges the front are smooth 63 Case III: χ = Consider the ase when the value of dimensionless parameter is very high ie and we find the values of parameters and suh that theoretial speed of growth front v is 1 Table 4 shows the values of dimensionless parameter for orresponding values of parameter In the modified Fisher Equation (10), growth and diffusion are taking plae simultaneously But if the diffusion is very small ompared to growth, then first ells will grow quikly and when they reah imum arrying apaity growth stops and they spread in the domain via diffusion In the present ase the growth term is very big as ompared to the diffusion term Figures 9 and 10 show the numerial solution of the modified Fisher s Equation (10) for and values of parameters and given in Table 4 The wave front takes some time to settle down to a travelling wave, whih moves at a onstant speed When the number of ells reahes its imum limit the proliferation Figure 7 Numerial results of profile of ell density N at different times when 1, , N init, Δt and t new are same as in Figure 5 In this ase the final time is t = 06 Figure 8 Numerial results of profile of ell density N at different times when, , N init, Δt and t new are same as in Figure 5 In this ase the final time is t = 06 Table 4 Values of and for stops and then the ells spread via diffusion in the entire domain In this ase diffusion term is muh smaller than the growth term, and due to this reason the shape of the front is very sharp It is evident from the Figure 9 and 10 that when the front settles down then it moves with Copyright 013 SiRes

10 M SHAKEEL 157 Figure 9 Numerial results of profile of ell density N at different times when 1, , N init, Δt and t new are same as in Figure 5 In this ase the final time is t = 06 Figure 11 Numerial results of profile of ell density N at different time without growth term when γ = 1, δ = , χ = 0 N init, Δt and t new are same as in Figure 5 Figure 10 Numerial results of profile of ell density N at different times when, , N init, Δt and t new are same as in Figure 5 In this ase the final time is t = 06 onstant speed and shape 64 Case IV: χ = 0 If 0 then it represents the ase when there is no growth of ells The number of ells will not inrease beause of the absene of growth term In that ase for every value of the solution will not evolve into a travelling wave Figures 11 and 1 show the ell density in the pure diffusion ase ie Fisher equation without the growth term for the same times and parameter values used in Figure 7 exept in this ase 0 The behaviour of the solution is different from the Fisher equation Figure 1 Numerial results of profile of ell density N at different time without growth term when γ =, δ = , χ = 0 N init, Δt and t new are same as in Figure 5 with the growth term Clear ly the solution does not grow due to the absene of growth term and the behaviour of the solution is not wave-like 7 Numerial Minimum Wave Speed From the phase plane analysis it is lear that a travelling wave front solution exists for a range of wave speeds v v We hoose the values of parameters and suh that the theoretially wave speed v 1 If di f- fusion is linear in the modified Fisher Equation ( 10) then travelling wave move with the minimum wave speed v v [] In this model 0 orresponds to linear Copyright 013 SiRes

11 158 M SHAKEEL diffusion But when the value of 0 then diffusion is no longer linear Here we see that the numerial wave speed agrees with the theoretial value when 3 However at 3 the numerial speed begins to diverge from the theoretial value and beome inreasingly large as inreases Figures 13 and 14 show the phase plane sketh of t he trajetories of Equation (10) for v v and v v respetively We observe from the Table 5 that when and 3 the numerial value of the minimum wave speed v mi n We see that from Figure 13 that when v 115 then all trajetories from 1,0 to 0,0 remain entirely in the region where 0 and 0 for all wave speed v 115 Simi- Figure 14 we observe that when wave speed larly from v 115 then for all the trajetories from 1,0 to 0,0, beomes negative, whih is unphysial 0,0 is still a stable node We observe that method of finding v by looking at the eigenvalues gives the wrong answer It is beyond the sope of this work to find the analytial formula for the minimum wave speed v min, beause the solution depends on the whole trajetory, but we have a very good agreement between numerial results and phase plane analysis Figure 14 Phase plane trajetories of Equation (4) for different values of v v The other parameter values areχ = , γ = 3, and δ = Eah urve (starting from bottom) represents the trajetory for various values of veloity v v ie v = 1, v = 10, v = 105 and v = 109 Table 5 Numerial minimum wave speed v min v v min min Figure 15 shows the shape of growth front at time t 03 for fixed value of but different values of and It is evident from Figure 15 that the speed of the grow th front is not the same for all values of parameter The speed of the growth front depends on the value of and inreases as value of inreases Figure 13 Phase plane traj etories of Equation (4) for different values of v v The other parameter values are χ = , γ = 3, and δ = Eah urve (starting from bottom) represents the trajetory for various values of veloity v v ie v = 115, v = 1, v = 13 and v = 15 8 Conlusion In this paper we studied the modified Fisher-Kolmogorov equation with non-linear diffusion, in detail The diffusion oeffiient in this ase is non-linear, depending on the ell density The form of non-linear diffusion is suh that it produes similar behaviour to ell proliferation Copyright 013 SiRes

12 M SHAKEEL 159 Figure 15 Shape of growth front at time t = 03 for fixe d and value of for orresponding value of are given in Table 3 N in it, Δt and t new are same as in Figure 5 The diffusion is high where the ell density is high and it is low in the regions of low ell density but it remains positive This leads us to hoose the non-linear diffusion to be an exponential funtion of ell density We found a travelling wave solution of the Fisher equation and found the theoretial minimum wave speed of growth front by using an eigenvalue analysis of stationary points The results reveal that for moderately non-linear diffusion the wave speed found by eigenvalue method agrees with the results of the wave speed found numerially The waves of this kind are alled Pulled front [38] However for highly non-linear diffusion, numerial minimum wave speed of growth front is greater than the minimum speed of growth front found by using the eigenvalue analysis The waves of this kind are alled Pushed front in whih the wave speed is determined by non-linear effets We tested the system for various values of parameters and found that the system exhibits the travelling wave solution when the growth term is dominant over the diffusion term We also found that the minimum wave speed of growth front depends on the values of parameter and it inreases as value of inreases The Fish er equation is widely used in modelling the physial phenomenon where diffusion and growth are taking plae simultaneously This equation is very useful in modelling the ell growth in in vitro tissue engineering, beause the form of diffusion used in this model produes the similar behaviour as ell proliferation REFERENCES [1] A Okubo, D iffusion and Eologial Problems: Matheematis matial Models, Springer-Verlag, Berlin, 1980 [] J D Murray, Mathematial Biology, Biomath Texts, Vol 19, Springer Verlag, New York, 1989 [3] J G Skellam, Random Dispersal in Theoretial Popula- tions, Bulletin of Mathematial Biology, Vol 53, No 1, 1991, pp [4] J A Sherratt and J D Murray, Models of Epidermal Wound Healing, Proeedings: Biologial Sienes, Vol 41, No 1300, 1990, pp 9-36 [5] M A J Chaplain and A M Stuart, A Model Mehanism for the Chemotati Response of Endothelial Cells to Tumour Angiogenesis Fator, Mathematial Mediine and Biology, Vol 10, No 3, 1993, pp [6] J A Sherratt, Wavefront Propagation in a Competition Equation with a New Motility Term Modelling Contat Inhibition between Cell Populations, Proeedings of the Royal Soiety of London Series A: Mathematial, Physial and Engineering Sienes, Vol 456, No 00, 000, pp [7] M A J Chaplain and A M Stuart, A Mathematial Model for the Diffusion of Tumour Angiogenesis Fator into the Surrounding Host Tissue, Mathematial Mediine and Biology, Vol 8, No 3, 1991, pp [8] M Aberrombie, Contat Inhibition in Tissue Culture, In Vitro Cellular & Developmental Biology-Plant, Vol 6, No, 1970, pp [9] P M Kareiva, Loal Movement in Herbivorous Insets: Applying a Passive Diffusion Model to Mark-Reapture Field Experiments, Oeologia, Vol 57, No 3, 1983, pp 3-37 [10] D Tilman and P M Kareiva, Spatial Eology: The Role of Spae in Population Dynamis and Interspeifi Interations, Prineton University Press, Prineton, 1997 [11] R A Fisher, The Wave of Advane of Advantageous Genes, Annals of Eugenis, Vol 7, No 4, 1937, pp [1] A Kolmogorov, I Petrovsy and N Piskounov, Study of the Diffusion Equation with Growth of the Quantity of Matter and Its Appliations to a Biologial Problem, Mosow University Mathematis Bulletin, 1989, p 105 [13] D A Frank-Kamenetskii, Diffusion and Heat Exhange in Chemial Kinetis, Prineton University Press, Prineton, 1955 [14] J Canosa, Diffusion in Non-Linear Multipliative Me- Diffusion Problems, In: A H dia, Journal of Mathematial Physis, Vol 10, No 10, 1969, pp [15] H Cohen, Non-Linear Taub, Ed, Studies in Applied Mathematis, The Mathematial Assoiation of Ameria, 1971, pp 7-64 [16] P C Fife and J B MLeod, The Approah of Solutions of Non-Linear Diffusion Equations to Travelling Front Solutions, Arhive for Rational Mehanis and Analysis, Vol 65, No 4, 1977, pp [17] J D Murray, Letures on Non-Linear Differential-Equation Models in Biology, Oxford University Press, Oxford, 1977 [18] H C Tukwell, Introdution to Theoretial Neurobiology: Non-Linear and Stohasti Theories, Cambridge University Press, Cambridge, 1988 Copyright 013 SiRes

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