Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions

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1 This article was downloaded by: [Texas Technology University] On: 23 April 2013, At: 07:52 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Stochastic Analysis and Applications Publication details, including instructions for authors and subscription information: Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions Ummugul Bulut a & Edward J. Allen a a Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, USA Version of record first published: 18 Jun To cite this article: Ummugul Bulut & Edward J. Allen (2012): Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions, Stochastic Analysis and Applications, 30:4, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Stochastic Analysis and Applications, 30: , 2012 Copyright Taylor & Francis Group, LLC ISSN print/ online DOI: / Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions UMMUGUL BULUT AND EDWARD J. ALLEN Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, USA 1. Introduction Stochastic partial differential equations for the one-dimensional telegraph equation and the two-dimensional linear transport equation are derived from basic principles. The telegraph equation and the linear transport equation are well-known correlated random walk (CRW) models, that is, transport models characterized by correlated successive-step orientations. In the present investigation, these deterministic CRW equations are generalized to stochastic CRW equations. To derive the stochastic CRW equations, the possible changes in direction and particle movement for a small time interval are carefully determined. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the position intervals decrease, stochastic partial differential equations are derived for the telegraph and transport equations. Comparisons between numerical solutions of the stochastic partial differential equations and independently formulated Monte Carlo calculations support the accuracy of the derivations. Keywords Stochastic differential equation; Stochastic partial differential equation; Telegraph equation; Transport equation. Mathematics Subject Classification 82C70; 60H15; 82C41; 60H10; 65C30. Stochastic differential equations (SDEs) are providing additional insight into randomly varying dynamical problems in mathematical biology, finance, engineering, chemistry, physics, and medicine [2, 3, 6, 9, 10, 13, 14]. Stochastic generalizations of deterministic models for certain dynamical systems are proving to be useful. In the present investigation, stochastic telegraph equations are derived in one-dimension and stochastic linear transport equations are derived in two-dimensions. Usually organisms and particles do not move in purely random directions. Often the current direction is correlated with the direction of prior movement. Received April 13, 2011; Accepted April 25, This work was partially supported by NSF grant DMS Address correspondence to Ummugul Bulut, Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas , USA; gul.bulut@ttu.edu 553

3 554 Bulut and Allen This type of random walk is called a correlated random walk (CRW) [11]. In the present investigation, two well-known correlated random walk models are studied, specifically, the telegraph equation in one-dimension and the linear transport equation in two dimensions. These equations are useful, for example, in studying animal or particle movement. In the present investigation, stochastic telegraph and linear transport equations are derived from basic principles. To derive an SPDE for a correlated random walk, all independent variables are made discrete. A discrete stochastic model is then developed by carefully studying the changes that occur in a small time interval. Next, letting the time interval go to zero yields an SDE system. Finally, the Wiener processes in the SDE system are appropriately replaced by Brownian sheets and the intervals in the remaining independent variables are allowed to go to zero. The resulting stochastic partial differential equation is an SPDE model for a correlated random walk. Before deriving stochastic generalizations of correlated random walk equations in one- and two-dimensions, it is useful to review some properties of Brownian sheets. Then, in the following two sections, a stochastic version of the telegraph equation is derived and a stochastic version of the two-dimensional linear transport equation is derived. Computational results are next described that illustrate that the stochastic correlated random walk derivations are reasonable. The results are summarized in the final section. 2. Some Properties of Brownian Sheets Before deriving the stochastic partial differential equations, it is useful to consider several properties of Brownian sheets [4, 8, 18]. A Brownian sheet on is illustrated in Figure 1. The Brownian sheet W t satisfies: t+ t x+ t x 2 W t dx N 0 t t Figure 1. A Brownian Sheet on (Figure available in color online.)

4 SPDEs for Correlated Random Walks 555 That is, the Brownian sheet is independent and normally distributed over rectangular regions. In addition, if x j = x min + j for j = 0 1 K, where = max x min /K, then the Brownian sheet defines for j = 1 2 K, the standard Wiener processes, W j t, where dwj t = xj x j 1 Notice that if t i = i t for t = 0 1 M, then 2 W t t dx ti t i 1 dw j t = t i j where i j N 0 1 for each j = 1 2 K and i = 1 2 M. In addition, from a two-dimensional Brownian sheet, an independent one-dimensional Wiener process in t can be defined for each x by W 1 x+ W t t x = lim dx 0 x In particular, W t x N 0 t for each t 0 and if x 1 x 2, then the Wiener process W t x 1 is independent of the Wiener process W t x 2. (Notice that W t x is not a Brownian sheet but is a one-dimensional Wiener process for each value of x.) These definitions can be extended to higher dimensions. For example, standard Wiener processes W i j t can be defined for j = 1 2 J and i = 1 2 I using a three dimensional Brownian sheet by 1 xi + dw i j t = y x i yj+ y y j 3 W y t t y where W y t is a three-dimensional Brownian sheet. dy dx 3. Derivation of Stochastic Correlated Random Walk Transport Models In this section, SPDEs are derived for the telegraph equation in one dimension and for the linear transport equation in two dimensions. In particular, the dynamical systems, with time discrete, are studied to determine the different independent random changes. As the time interval decreases, the discrete stochastic models lead to certain stochastic differential equation systems. Then, Brownian sheets are appropriately substituted for Wiener processes in SDE systems. When intervals in the secondary variables go to zero, the final SPDE models are derived Telegraph Equation for One-Dimensional Correlated Random Walk In this section, an SPDE is derived for a one-dimensional correlated random walk, specifically, the telegraph equation. The deterministic telegraph equation is first introduced and described before deriving a stochastic generalization. The deterministic telegraph equation is derived by considering right and left moving particles. Let t and t be the number densities of right and left moving particles at position x and time t. In particular, and have units of

5 556 Bulut and Allen number of particles per unit distance. The rate of change for the right and left moving particles can be shown to satisfy the equations: = v + t (3.1) t = v (3.2) where v is speed and is the rate that the particles change direction. Adding (3.1) and (3.2) and differentiating with respect to t results in the equation: 2 + t 2 = v 2 (3.3) t Subtracting (3.1) from (3.2) and differentiating with respect to x gives 2 t = v (3.4) Finally, substituting (3.4) into (3.3) and using (3.1) and (3.2), the well-known telegraph equation is obtained: 2 p t 2 p + 2 t = 2 p v2 (3.5) 2 where p = p t = t + t is the total particle density. The telegraph Equation (3.5) has some similarities with the simple diffusion process. Dispersal of populations or particles is often modeled using a diffusion process. Researchers, however, sometimes criticize diffusion models of population movement since some assumptions in the diffusion process are unrealistic such as infinite velocity of organism movement. An interesting discussion of the comparison of the diffusion process with telegraph transport is found in [15]. The telegraph equation has distinctive properties depending on. For small, solutions to the telegraph equation behave in a wave-like manner and for large, solutions to the telegraph equation behave in a diffusion-like manner. This behavior of the telegraph equation is illustrated in Figures 2 and 3. In these figures, Equation (3.5) is solved on 1 1 up to time 0.5 and speed unity with the initial density equal to 1 on 1 1 and 0 otherwise. The value of is 1.5 for Figure 2 and the value of the is 7 for Figure 3. The transport clearly behaves wave-like when = 1 5 and behaves diffusion-like when = 7. A stochastic version of the telegraph equation is now derived. In telegraph movement, particles (microorganism, bacteria, etc.) move either right or left with a constant velocity v. In the present investigation, it is assumed that particles have a possible change in direction with probability per unit distance. When a possible change in direction occurs, the probability of going straight is 1. Thus, is the probability that a particle changes direction after an interaction. Notice that for a biological population model, may be considered as a decision parameter and for a physical particle model, may be a scattering cross section.

6 SPDEs for Correlated Random Walks 557 Figure 2. The solution of the telegraph equation on 1 1 at time 0.5 for = 1 5 with initial condition p 0 =1 on 1 1. Let t and t be the population densities per unit distance of right moving particles and left moving particles, respectively. Consider now the changes that occur in the right and left moving particle populations for a small time interval t. To find these changes, the spatial interval min max is divided into N subintervals i 1 i, for i = 1 2 N where x 0 = x min, x N = x max, x j = x min + j for j = 1 2 N 1, and = max x min /N. Let i t and i t be the right and left moving particle population levels on the interval i 1 i at time t. The changes possible for i t and i t for a small time interval, t, are listed in Tables 1 and 2. These tables define a discrete stochastic model for telegraphic movement in one-dimension. Consider the possible changes with their respective probabilities for right moving particles on the ith interval, i 1 i, as listed in Table 1. Since we assumed that particles are moving continuously with a constant velocity v, i 1 t v t right moving particles pass into the ith interval from the previous interval in time interval t with probability unity. Similarly, i t v t right moving particles cross into the next interval in time t with probability unity. Also, one right moving particle on the ith interval changes direction with probability i t v t. Figure 3. The solution of the telegraph equation on 1 1 at time 0.5 for = 7 with initial condition p 0 =1 on 1 1.

7 558 Bulut and Allen Table 1 Right-moving particle population changes on i 1 i in time t Possible change i Probability i 1 v t/ p 1 = 1 i v t/ p 2 = 1 0 p 3 = i v t 1 1 p 4 = i v t 1 p 5 = i v t In addition, one left-moving particle changes direction and becomes a right-moving particle with probability i t v t. The random part of the derived stochastic transport equation for the right moving particle population arises from the last two terms in Table 1. The changes and probabilities in Table 2 are constructed similarly as those in Table 1. Tables 1 and 2 define a discrete stochastic model for telegraphic movement in one-dimension. A certain SDE system approximately satisfies, for small t, the same probability distribution as the discrete stochastic model defined by Tables 1 and 2. For a discussion of the similarities in probability distributions for discrete and continuous stochastic models, see, for example, references [1, 3, 5, 6]. Based on the changes and probabilities in Tables 1 and 2, the particle population levels i t and i t, for i = 1 2 N, approximately satisfy the following SDE system: d i t d i t = i 1 t i t v i t v d W i t = i+1 t i t v i t v dw i t i t v + i t v + i t v dw i t i t v + i t v + i t v d W i t (3.6) (3.7) Table 2 Left-moving particle population changes i 1 i in time t Possible change i Probability i+1 v t/ p 1 = 1 i v t/ p 2 = 1 0 p 3 = i v t 1 1 p 4 = i v t 1 p 5 = i v t

8 SPDEs for Correlated Random Walks 559 where W i t, W i t are independent Wiener processes for i = 1 N. Moreover, the Euler-Maruyama approximation to this SDE system is useful and is given by: i t + t i t + i 1 t v t/ i t v t/ i t v t + i t v t i t v t i + i t v t i (3.8) i t + t i t + i+1 t v t/ i t v t/ i t v t + i t v t i t v t i + i t v t i (3.9) where i, i, are independent normally distributed numbers with mean zero and variance unity for i = 1 N. To derive SPDEs from systems (3.6) and (3.7), i t and i t are replaced by i t and i t, respectively, where t and t are the right and left moving particle population densities at position x and time t. Then, these equations are divided by to obtain i t t i t t = i 1 t t i t v i t v + i t v i t v d W i t i t v dw + i t = i+1 t i t v i t v + i t v i t v dw i t i t v d W + i t (3.10) (3.11) The Wiener processes are replaced with suitable Brownian sheets, and as approaches zero, the following SPDEs are derived: t t t t v t = t v + t v = t v 2 W t x t v t t v + t v t v 2 W t x + t v 2 W t x t + t v 2 W t x t (3.12) (3.13) where W t x and W t x are independent two-dimensional Brownian sheets. In particular, dw i t = 1 xi x i 1 2 W t x t dx for i = 1 2 N

9 560 Bulut and Allen and d W i t = 1 xi x i 1 2 W t x t dx for i = 1 2 N Equations (3.12) and (3.13) are stochastic telegraph equations in one dimension and generalize the telegraphic system (3.1) (3.2). Notice that in (3.1) and (3.2) is replaced with = v in (3.12) and (3.13). In the present investigation, a stochastic version of the telegraph Equation (3.5) for p t is not formulated and studied. In particular, to calculate the total particle density p t = t + t for stochastic telegraphic transport, Equations (3.12) and (3.13) are independently solved and their solutions are summed Transport Equation in Two Dimensions The telegraph process is a correlated random walk in one dimension. In two dimensions, linear transport may be considered a generalized version of a telegraphic process. The linear transport equation is a correlated random walk model and is useful, for example, in describing the movement or reorientation of cells and animals [7]. In this two-dimensional transport model, it is assumed that particles with a constant velocity are moving on a plane in a direction where < <. Particles change direction from direction k to direction k with probability s k k per unit distance and per unit angle. Let N y t be the number of particles at position y and direction per unit area and per unit angle at time t. Thus, for example, the integral 2 x2 y2 1 x 1 y 1 N y t dydxd gives the total number of particles on rectangle 1 2 y 1 y 2 moving in direction, for between 1 and 2. To derive a stochastic generalization of the linear transport equation, twodimensional space is made discrete by dividing min max y min y max into I and J equal intervals respectively. Let = x i+1 x i, for i = I 1, and y = y j+1 y j, for j = J 1. Also, direction interval is discretized with K possible directions k = + k 5 that particles move for k = 1 2 K where = 2 /K. Before determining the changes that occur in each rectangular region for a small interval of time, some notation is first introduced. Define n ijk = N i y j k t y. Thus, n ijk is the total number of individuals moving in direction k in region i i+1 y j y j+1 at time t. Let k k = s k k be the probability of changing direction from k to k per unit distance and per unit angle. The changes possible along with their probabilities for n ijk are listed in Table 3 for a direction k between 0 and /2 with the transport illustrated in Figure 4. In Table 3, the changes are listed as n ijk along with their corresponded probabilities. Notice that, for example, n i 1jk cos k v t particles, moving in direction k, pass from rectangle i 1 i y j y j+1 into rectangle i i+1 y j y j+1 with probability unity. Similarly, particles are entering or leaving the rectangle i i+1 y j y j+1 in direction k from other border rectangles. The first line in Table 3 is a gain of one particle in direction k due to a scatter from a particle in direction k, where k k. The second line in Table 3 denotes a loss of one particle from direction k into direction k due to a scatter. Using the changes and probabilities in Table 3 where

10 SPDEs for Correlated Random Walks 561 Table 3 Changes and probabilities for the number of particles moving in direction k in rectangle i i+1 y j y j+1 for two-dimensional transport Possible change n ijk Probability +1 p 1 = n ijk k k t 1 p 2 = n ijk kk t +n i 1jk cos k t/ p 3 = 1 n ijk cos k t/ p 4 = 1 n ij 1k sin k t/ y p 5 = 1 n ijk sin k t/ y p 6 = 1 0 < k < /2 along with analogous changes and probabilities for other directions k, the following SDE system is obtained: dn ijk t = n ijk t kk + n ijk t k k + n i 1jk t cos k n ijk t cos k 1 + n ij 1k t sin k n ijk t sin k 1 y n ijk kk dw 1/2 ijkk t + n ijk k k dw 1/2 ijk k t (3.14) where W ijkk t and W ijk k t are independent Wiener processes with mean 0 and variance unity, for i = 0 I 1, j = 0 J 1 and k = 1 K. System (3.14) has approximately the same probability distribution as the discrete stochastic model defined by the changes and probabilities given in Table 3. Figure 4. Particle number n ijk for one specific direction on a two-dimensional mesh for linear transport.

11 562 Bulut and Allen An approximation to the SDE system for small t is useful and is given by: n ijk t + t n ijk t n ijk t kk t + n ijk t k k t + n i 1jk t cos k t n ijk t cos k t + n ij 1k t sin k t y n ijk t sin k t K y n ijk kk t 1/2 ijkk + n ijk k k t 1/2 ijk k (3.15) where ijkk and ijk k are normally distributed random numbers with mean zero and variance unity for i = I 1, j = J 1, and k = 1 2 K. To derive an SPDE, SDE system (3.14) is divided by y and n ijk t / y is replaced by N i y j k t. Next, Wiener processes are replaced with a suitable five-dimensional Brownian sheet. As, y, 0, the following SPDE is derived: N y t t = vcos N where N v sin y + N y t v s d + dw ijkk t = N y t v s d N y t v s 5 W y t y t d N y t v s 5 W y t d (3.16) y t 1 k +1 k+1 yj+1 xi+1 y k k y j x i 5 W y t dx dy d d y t Equation (3.16) is the stochastic linear transport equation in two-dimensions. 4. Comparison with Monte Carlo Calculations In this section, the derived stochastic partial differential equations for correlated random walks in one- and two-dimensions are solved numerically and tested against independent Monte Carlo calculations. One computational example is considered for each SPDE. In a correlated random walk process, there may be either a fixed or variable step length between direction changes. In the present work, a fixed step length is used in the computational examples. The first problem is one-dimensional. It is assumed that there are 1800 right and left moving particles on 4 4. The right and left moving particles are distributed identically. The density of the particles is 100 on , 1000x on 1 0, and x on 0 1. Initially, there are no particles outside the

12 SPDEs for Correlated Random Walks 563 interval 4 4. Particles move with a constant velocity v = 1. The probability of having an interaction per unit distance is = 1. After an interaction occurs, the probability of changing direction is = 0 5. The particle distribution on the interval 4 4 is observed at time 0.5. The spatial interval 4 4 is divided into I equal subintervals i 1 i, for i = 1 2 I where x i = i, and I = The problem is solved computationally using a numerical approximation to the stochastic telegraph Equations (3.12) and (3.13). Specifically, the Euler-Maruyama approximation is used which has the form: i t + t i t i 1 t v t/ i t v t/ i t v t + i t v t i t v t i + i t v t i (4.17) i t + t i t i+1 t v t/ i t v t/ i t v t + i t v t i t v t i + i t v t i (4.18) where i t i t and i t i t are the number of right and left moving particles at position x i at time t. Also, i, i, are independent normally distributed numbers with mean zero and variance unity for i = 1 N. Notice that (4.17) and (4.18) are Euler-Maruyama approximations [12, 16, 17] to the systems of Itô differential Equations (3.6) and (3.7). The problem is solved numerically using two independent computational procedures, i.e, numerical solution of the SPDE is compared with Monte Carlo calculations. Equations (4.17) and (4.18) are solved computationally with I = 1280 equal intervals in position x. The value chosen for the time interval is t = 1/200. It is assumed that there exists a vacuum outside 4 4 with no entering particles at 4 and 4. In the Monte Carlo procedure, each particle is followed individually with each particle checked for an interaction or direction change at each time step. Calculational results for 500 sample paths using the two independent computational approaches are given in Table 4. In the table, the mean and standard deviation (Sd) of the number of particles in interval 1 1 at times , 0.125, 0.25, and 0.5 are given for the numerical solution of the SPDE and the Monte Carlo calculations. In Figures 5 and 6, the mean and mean-squared values of SPDE and Monte-Carlo results are given in the interval [ 1,1] for time = 25. The two different computational approaches agree well. The second problem is two dimensional. It is assumed that initially there are 1024 particles with constant velocity = 1 uniformly distributed on a unit square Outside the unit square is a vacuum. The interval 0 2 is divided Table 4 One-dimensional Monte Carlo (MC) and SPDE calculational results for 500 sample paths Time Mean (SDE) Mean (MC) Time Sd (SDE) Sd (MC)

13 564 Bulut and Allen Figure 5. Mean number of particles on 1 1 at time 0.25 for the SPDE and MC. (Figure available in color online.) into 16 particle directions. Isotropic scattering is assumed and the probability of changing direction from k to k is taken as k k = s k k = The particle distribution on the square is observed from time 0 until 1.6. The unit square is divided into 256 smaller squares. The problem is solved computationally using a numerical solution to SPDE (3.16). In particular, an Euler-Maruyama approximation to the SDE system (3.14) is used which has the form: n ijk t + t n ijk t n ijk t kk t + n ijk t k k t + n i 1jk t cos k t n ijk t cos k t + n ij 1k t sin k t y n ijk t sin k t K y n ijk kk t 1/2 ijkk + n ijk k k t 1/2 ijk k (4.19) Figure 6. Standard deviations in the number of particles on 1 1 at time 0.25 for the SPDE and MC. (Figure available in color online.)

14 SPDEs for Correlated Random Walks 565 Table 5 Two-dimensional Monte Carlo (MC) and SPDE calculational results for 1000 sample paths Method Mean Sd SPDE MC Figure 7. Mean number of particles in from time 0 to time 1.6 for the SPDE and MC. where n ijk t is the number of particles in direction k at time t. Also, ijkk and ijk k are normally distributed numbers with mean 0 and variance unity for i = I 1, j = J 1, and k = 1 2 K. Figure 8. Standard deviation in number of particles in from time 0 to time 1.6 for the SPDE and MC.

15 566 Bulut and Allen This problem is solved numerically by using two procedures, i.e, numerical solution of the SPDE is compared with Monte Carlo calculations. Equation (4.19) is solved computationally with I = 16 equal intervals in position x and y. The value chosen for the time interval is t = 1/160. In the Monte Carlo procedure, each particle is followed individually with each particle checked for a direction change at each time step. Both spatial and direction variables are continuous in the Monte Carlo procedure, that is, particles are independently followed in space and direction. Calculational results for 1000 sample paths using the two independent computational approaches are given in Table 5 for time t = 1 6. The means and standard deviations of the calculated numbers of particles in the unit square are given at time 1.6. In Figures 7 and 8, the mean and mean-squared values of the SPDE and Monte-Carlo results are given up to time = 1 6. The two different computational approaches agree well. 5. Summary and Conclusions In this study, stochastic partial differential equations are derived for correlated random walk models in one and two dimensions. The SPDEs are solved numerically and compared with independently formulated Monte Carlo methods. The computational results between the two different numerical methods are in good agreement supporting the accuracy of the derived SPDEs. References 1. Allen, E.J Derivation of stochastic partial differential equations. Stoch. Anal. Appl. 26: Allen, E.J Derivation of stochastic partial differential equations for size- and age-structured populations. Journal of Biological Dynamics 3: Allen, E.J Modeling With Iô Stochastic Differential Equations. Springer, Dordrecht. 4. Allen, E.J., Novosel, S.J., and Zhang, Z Finite element and difference approximation ofsome linear stochastic partial differential equations. Stochastics and Stochastics Reports 64: Allen, E.J., Allen, L.J.S., Arciniega, A., and Greenwood, P.E Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl. 26: Allen, L.J.S An Introduction to Stochastic Processes with Applications to Biology. Pearson Education, Upper Saddle River, NJ. 7. Alt, W Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9: Cabaña, E.M The vibrating string forced by white noise. Zeitschriftfür Wahrscheinlichkeitstheorie und verwane Gebiete 15: Capasso, V., and Bakstein, D An Introduction to Continuous-Time Stochastic Processes, Birkhäuser, Boston. 10. Da Prato, G., and Tubaro, L. (Eds.) Stochastic Partial Differential Equations and Applications VII. CRC Press, Boca Raton, FL. 11. Codling, E.A., Plank, M.J., and Benhamou, S Random walk models in biology. J. R. Soc. Interface 5: Gard, T.C Introduction to Stochastic Differential Equations. Marcel Decker, New York. 13. Gunzburger, M Numerical methods for stochastic PDEs. SIAM News 40: Holden, H., Øksendal, B., Ubøe, J., and Zhang, T Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach. Birhäuser, Boston.

16 SPDEs for Correlated Random Walks Holmes, E.E Are diffusion models too simple? A Comparision with telegraph models of invasion. The American Naturalist 142: Kloeden, P.E., and Platen, E Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York. 17. Kloeden, P.E., Platen, E., and Schurz, H Numerical Solution of SDE Through Computer Experiments. Springer, Berlin. 18. Walsh, J.B An introduction to stochastic partial differential equations. In Lecture Notes in Mathematics. Dold, A. and Eckmann, B. (eds.), Vol. 1180, Springer-Verlag, Berlin, pp