Biased Random Walks in Biology

Transcription

1 Biased Random Walks in Biology Edward Alexander Codling Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds, Department of Applied Mathematics. August 2003 The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgment.

2 Acknowledgements I would like to thank my supervisor Prof. Nick Hill for his calm guidance and unfailing help and support throughout four years of research. He has always made time to help me and has supported all my activities, whether part of my research or not. His enthusiasm for the subject has kept me inspired throughout and I feel privileged to have been able to share in his expert knowledge through the many discussions we have had. While at Leeds, I have had many helpful and rewarding discussions with everyone involved with the Biomaths group, in particular Prof. Brian Sleeman and Mike Plank. I have enjoyed working with Dr. Jon Pitchford, who has been very encouraging and eager to help with the fish larvae project, and always keen to discuss any other aspect of the rest of my research. Dr. Steve Simpson has been extremely helpful in passing on his detailed and expert knowledge of fish larvae behaviour. I am grateful for the generous assistance of Chris Needham in helping me to program in C and C++ without such help I would not have been able to set up and run such detailed numerical simulations. Finally, I must give special thanks to the Carswell family: Neil, Eleanor, Annie and Thomas, for supporting and accommodating me during the writing up period, and also the rest of my family and friends (especially Becky) for all the help, support, and biscuits. Financial support for this research has been provided by the E.P.S.R.C. i

3 Abstract Random walks are used to describe the trajectories of many motile animals and microorganisms. They are a useful tool for both qualitative and quantitative descriptions of the behaviour of such creatures. Simple diffusive random walk models, or position jump processes, are unrealistic as they allow for effectively infinite propagation and do not take into account correlation between steps. Othmer et al. (1988) use a generalised transport equation to model biased and correlated velocity jump processes where the speed of movement is finite. In two dimensions an equation for the underlying spatial distribution of the velocity jump process model cannot be found, so Othmer et al. use a method of calculating moments to derive and solve differential equations for the statistics of interest. We extend the velocity jump process and method of moments used by Othmer et al. to include reorientation models where the mean turning angle is dependent on the previous direction of movement, as observed by Hill & Häder (1997) in experiments on algae. Closure assumptions are made in order to derive and solve a system of differential equations for the higher order moments and statistics of the underlying spatial distribution. Numerical simulations are then used to compare the asymptotic solutions with simulated data, and the fit is good for biologically realistic parameter values. Numerical simulations of velocity jump processes are also used to investigate the method used by Hill & Häder to calculate the reorientation parameters from the angular statistics of a random walk, and also to investigate the effect of spatially dependent parameters or a changing preferred direction on the spatial statistics. We look at the ratio between the root of the mean squared displacement and the mean dispersal distance in both unbiased and biased random walks and demonstrate how this can give us more information about the spatial distribution. We give an application of the velocity jump process model to the movement and recruitment of reef fish larvae. The variability in the movement is found to be important if there is a low survival probability, while in simple reef environments the survival probability appears to be highly sensitive to the reorientation parameters and corresponding swimming behaviour. ii

4 Contents 1 Introduction and background General background to random walks History The isotropic random walk and the diffusion equation Random walks to a barrier a simple example The telegraph equation Circular statistics The mean direction The mean resultant length and the circular variance Probability distributions on the circle Modelling biological motion The movement of animals and micro-organisms as a random walk Biased movement and taxis Other applications of the random walk in biology Properties of correlated random walks Mean squared displacement Sinuosity and mean dispersal distance The circular random walk and reorientation models arising from experiments on algae Deriving the Fokker Planck equation for a circular random walk Reorientation models and solutions to the Fokker Planck equation Experimental results Overview of subsequent chapters Simple two-dimensional random walk models Introduction Two-dimensional uncorrelated random walks Lattice model Turning probabilities independent of position Turning probabilities dependent on position iii

5 iv 2.4 Multi-directional discrete direction model and continuous direction model Multi-directional discrete direction model Continuous direction model Solution of the Fokker Planck diffusion equation Solution for isotropic movement Solution for biased movement The telegraph equation in higher dimensions Conclusions Spatial statistics of two-dimensional velocity jump processes Introduction Generalized equation for velocity jump processes Generalized model Velocity jump processes in one dimension the telegraph equation Velocity jump processes in two dimensions random walks in external fields Defining statistics of interest Deriving equations for spatial statistics Solving equations for spatial statistics Conclusions Velocity jump processes using sinusoidal reorientation Introduction Reorientation model Hill & Häder s general reorientation model The reorientation kernel T(θ,θ ) Sinusoidal reorientation model The biological relevance of the turning angle distribution parameters Defining statistics of interest Results and assumptions to be used in analysis Integrals of the von Mises distribution Asymptotic expansions of the trigonometric functions Previous results Other assumptions Differential equations for the spatial statistics and higher order moments Deriving equations for spatial statistics Deriving equations for the higher order moments Closing and solving the system of equations for H(t), V(t), F n (t) and Y n (t) Approximating the higher order moments The general solution to a linear system of differential equations Solving the final system of equations for H(t), V(t), F n (t) and Y n (t) 80

6 v 4.7 Closing and solving the system of equations for D 2 (t), G n (t) and Z n (t) Approximating the higher order moments Solving the final system of equations for D 2 (t), G n (t) and Z n (t) Equations for the spread about the mean position Solution plots Comment on solutions Working with the equations for the statistics of interest Limitations of the model and solutions Rescaling the equations Limits on the parameters Conclusions Velocity jump processes using linear reorientation Introduction Results and assumptions to be used in analysis Reorientation model Defining higher order moments Integrals of the von Mises distribution Asymptotic expansions of the trigonometric functions Previous results Differential equations for the spatial statistics and higher order moments Deriving equations for spatial statistics Deriving equations for the higher order moments System of equations for non-spatial moments System of equations for spatial moments Solving the systems of equations Solving for the non-spatial higher order moments Solving for V(t) and H(t) Solving for the spatial higher order moments Solving for D 2 (t) and σ 2 (t) Final system of solutions Numerical solutions Solution plots Comment on solutions Limitations of the model and solutions Comparing solutions of the sinusoidal and linear models Comparing solutions for H(t) Comparing solutions for σ 2 (t) Conclusions

7 vi 6 Spatial statistics of simulated random walks Introduction Computer simulations of random walks Simulation of an individual random walk Collecting average statistics for a set of random walks Simulations to validate theoretical results Mean position H(t) Average velocity V(t) Measure of spread about the origin D 2 (t) Measure of spread about the mean position σ 2 (t) The effect of the reorientation parameters on fixed time solutions Fixed time spatial distribution The effect of changing the reorientation parameters κ and d τ Simulations with parameters from experimental data Data set C1 (Sinusoidal model) Data set C3 (Linear model) Data set C4 (Linear model) Conclusions Angular statistics and the effect of sampling length Introduction The long-time absolute angular distribution Validating the approximation for M 0 (t) Comparing theoretical distributions to simulation results Moments of the long-time absolute angular distribution The effect of sampling length on the angular statistics of a velocity jump process Examples of changing the sampling length Angular statistics of a velocity jump process with sinusoidal reorientation Angular statistics of a velocity jump process with linear reorientation Limitations of using the angular statistics to estimate the reorientation parameters of a velocity jump process The effect of sampling length on the angular statistics of a velocity jump process with a fixed time between turns Estimating the reorientation parameters for large and small values of τ and τ s Conclusions

8 vii 8 Further modelling with computer simulations Introduction Simulations with parameter values outside the limits of the theoretical models The effect of the parameter κ The effect of the parameter d τ Theoretical optimal value of d τ Biological relevance of larger reorientation parameter values Simulations with non-constant parameters Spatial dependence of κ Spatial dependence of d τ Biological relevance of spatially dependent reorientation parameters Simulations with a changing preferred direction Reorientation models for a changing preferred direction Examples of individual random walks Average position H y (t) Spread about the mean position σ 2 (t) Conclusions Mean dispersal distance of correlated random walks Introduction The mean squared displacement Comparing the mean squared displacement for unbiased discrete random walks and velocity jump processes Mean squared displacement for variable and fixed step lengths The mean dispersal distance of unbiased random walks Calculating the mean dispersal distance from the mean squared displacement in a discrete random walk A better model for MDD(n) The mean dispersal distance of an unbiased velocity jump process with a variable time step The mean dispersal distance in each direction for an unbiased velocity jump process The mean dispersal distance of biased random walks The limiting value of the correction factor Simulated behaviour of the limiting value of the correction factor Conclusions Random walks to a barrier and the recruitment of fish larvae Introduction Background to fish larval movement and recruitment

9 viii Recruitment of fish larvae in the open sea Recruitment of reef fish larvae Theoretical models of fish larvae returning to a reef Experimental data for fish larvae returning to a reef The effect of variability on fish larvae recruitment Model 1: simple reef environment Deterministic model for population dynamics Stochastic model for population dynamics Survival probabilities for the simple reef model Optimal swimming behaviour for fish larvae attempting to recruit to a reef Model 2: simple circular reef model Model 3: simple current model Further models Conclusions Concluding remarks Main results Possible future research Bibliography 314

10 List of Figures 1.1 Plots showing P(x,t) for D = 1, 5 and 10, and t = 1 and 10. (The scales on each plot are different) Plots showing P(x,y,t) for D = 1 and 5, and t = 1 and 10. (The scales on each plot are different) Plots of P(x,t), the solution of the drift diffusion equation, for various u Plots of p(x, t), the solution of the telegraph equation for various parameter values Example of a data set on a circle with R 0 but with a non-uniform spread of points Examples of the von Mises distribution for various values of κ, and µ = Plot of κ against σ Plot comparing µ 0 (θ) for sinusoidal ( ) and linear reorientation (- -), for θ < π and B 1 = Example of a two-dimensional lattice random walk Example of a multi-directional random walk Plots showing f(x, y, t) for various parameter values at t = Sketch of the probability distributions for h(δ) and k(θ) as used by Othmer et al. (1988) Sketch showing the difference between D 2 and σ 2 (H is the average position) Plots of V x1 (t) and H x1 (t) for various values of C I Plots of D 2 (t) and σ 2 (t) for various values of C I Plot of V(t) for d τ = 0.1 and d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of H(t) for d τ = 0.1 and d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of D 2 (t) for d τ = 0.1 and d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of σ 2 (t) for d τ = 0.1 and d τ = 0.3 and various values of κ. (The scale of each plot is different) ix

11 x 4.5 Plot of Dx1 2 (t) and D2 x2 (t) for d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of σx1 2 (t) and σ2 x2 (t) for d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of ζ 1 against κ for d τ = 0,0.1,0.2, Plot of ζ 2 against κ for d τ = 0.1,0.2, Plots comparing k 1 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plots comparing l 1 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plots comparing m 1 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plots comparing n 1 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plots comparing k 2 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plots comparing l 2 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plots comparing m 2 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plots comparing n 2 (µ,κ) to the exact integral for various values of κ. (The scale of each plot is different) Plot of V(t) for d τ = 0.1 and d τ = 0.3 and various values of κ Plot of H(t) for d τ = 0.1 and d τ = 0.3 and various values of κ Plot of D 2 (t) for d τ = 0.1 and d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of σ 2 (t) for d τ = 0.1 and d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of Dx1 2 (t) and D2 x2 (t) for d τ = 0.3 and various values of κ. (The scale of each plot is different) Plot of σx1 2 (t) and σ2 x2 (t) for d τ = 0.3 and various values of κ. (The scale of each plot is different) Simple algorithm for an individual random walk i) Random walk with κ = 0.1, d τ = 0. The random walk is close to being completely random (Brownian) motion ii) Random walk with κ = 2, d τ = 0. The random walk appears more correlated but there is no overall preferred direction

12 xi 6.4 iii) Random walk with κ = 0.5, d τ = 0.2. The random walk is less correlated but there is a definite preferred direction (y-direction) iv) Random walk with κ = 4, d τ = 0.3. The random walk is highly correlated and the preferred direction is clear Algorithm used to calculate average statistics for a set of random walks Plots showing theoretical H y (t) ( ), and 95% confidence interval from simulated ( ), against time for sinusoidal reorientation with d τ = 0.1. (The scale used for each plot is different.) Plots showing theoretical H y (t) ( ), and 95% confidence interval from simulated ( ), against time for sinusoidal reorientation with d τ = 0.2. (The scale used for each plot is different.) Plots showing theoretical H y (t) ( ), and 95% confidence interval from simulated ( ), against time for sinusoidal reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing theoretical H y (t) ( ), and 95% confidence interval from simulated ( ), against time for linear reorientation with d τ = 0.1. (The scale used for each plot is different.) Plots showing theoretical H y (t) ( ), and 95% confidence interval from simulated ( ), against time for linear reorientation with d τ = 0.2. (The scale used for each plot is different.) Plots showing theoretical H y (t) ( ), and 95% confidence interval from simulated ( ), against time for linear reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), absolute velocity (H y (t)/t) in the y-direction against time for sinusoidal reorientation with d τ = 0.1. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ) absolute velocity (H y (t)/t) in the y-direction against time for sinusoidal reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), absolute velocity (H y (t)/t) in the y-direction against time for linear reorientation with d τ = 0.1. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), absolute velocity (H y (t)/t) in the y-direction against time for linear reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), D 2 (t) against time for sinusoidal reorientation with d τ = 0.1. (The scale used for each plot is different.)

13 xii 6.18 Plots showing theoretical ( ), and simulated ( ), D 2 (t) against time for sinusoidal reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), D 2 (t) against time for linear reorientation with d τ = 0.1. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), D 2 (t) against time for linear reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing D 2 x(t) and D 2 y(t) against time for sinusoidal reorientation with various values of the parameters. (The scale used for each plot is different.) Plots showing Dx 2(t) and D2 y (t) against time for linear reorientation with various values of the parameters. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), σ 2 (t) against time for sinusoidal reorientation with d τ = 0. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), σ 2 (t) against time for sinusoidal reorientation with d τ = 0.1. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), σ 2 (t) against time for sinusoidal reorientation with d τ = 0.2. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), σ 2 (t) against time for sinusoidal reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), σ 2 (t) against time for linear reorientation with d τ = 0.1. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), σ 2 (t) against time for linear reorientation with d τ = 0.2. (The scale used for each plot is different.) Plots showing theoretical ( ), and simulated ( ), σ 2 (t) against time for linear reorientation with d τ = 0.3. (The scale used for each plot is different.) Plots showing σx 2 (t) against time for sinusoidal reorientation with various values of the parameters. (The scale used for each plot is different.) Plots showing σ 2 y(t) against time for sinusoidal reorientation with various values of the parameters. (The scale used for each plot is different.) Plots showing σx 2 (t) against time for linear reorientation with various values of the parameters. (The scale used for each plot is different.) Plots showing σ 2 y(t) against time for linear reorientation with various values of the parameters. (The scale used for each plot is different.) Example plots of the population position and spread at t = Plots showing H y (100) against κ for sinusoidal and linear reorientation with d τ = 0 ( ), d τ = 0.1 ( ), d τ = 0.2 ( ) and d τ = 0.3 ( )

14 xiii 6.36 Plots showing D 2 (100) against κ for sinusoidal and linear reorientation with d τ = 0 ( ), d τ = 0.1 ( ), d τ = 0.2 ( ) and d τ = 0.3 ( ) Plots showing D 2 x(100) against κ for sinusoidal and linear reorientation with d τ = 0 ( ), d τ = 0.1 ( ), d τ = 0.2 ( ) and d τ = 0.3 ( ) Plots showing Dy 2 (100) against κ for sinusoidal and linear reorientation with d τ = 0 ( ), d τ = 0.1 ( ), d τ = 0.2 ( ) and d τ = 0.3 ( ) Plots showing σ 2 (100) against κ for sinusoidal and linear reorientation with d τ = 0 ( ), d τ = 0.1 ( ), d τ = 0.2 ( ) and d τ = 0.3 ( ) Plots showing σ 2 y(100) against κ for sinusoidal and linear reorientation with d τ = 0 ( ), d τ = 0.1 ( ), d τ = 0.2 ( ) and d τ = 0.3 ( ) Plots showing (a) final position and spread (t = 100), (b) H y (t), (c) σ 2 x (t) and (d) σ 2 y(t) for reorientation parameters from data set C1:a Plots showing (a) final position and spread (t = 100), (b) H y (t), (c) σ 2 x(t) and (d) σy 2 (t) for reorientation parameters from data set C1:b Plots showing (a) final position and spread (t = 100), (b) H y (t), (c) σx 2(t) and (d) σy 2 (t) for reorientation parameters from data set C3:a Plots showing (a) final position and spread (t = 100), (b) H y (t), (c) σx(t) 2 and (d) σ 2 y(t) for reorientation parameters from data set C3:b Plots showing (a) final position and spread (t = 100), (b) H y (t), (c) σ 2 x (t) and (d) σy 2 (t) for reorientation parameters from data set C4:a Plots showing (a) final position and spread (t = 100), (b) H y (t), (c) σx(t) 2 and (d) σ 2 y(t) for reorientation parameters from data set C4:b Plots of M 0 (t) against t. Legend: (- -) simulation κ = 1, ( ) simulation κ = 2, ( ) simulation κ = 4, (+) approximation κ = 1, (*) approximation κ = 2, ( ) approximation κ = Plots showing theoretical and simulated long-time p.d.f., f(θ), with parameter values taken from Hill and Häder s experiments with data set C Plots showing theoretical and simulated long-time p.d.f., f(θ), with parameter values taken from Hill and Häder s experiments with data set C Plots showing theoretical and simulated long-time p.d.f., f(θ), with parameter values taken from Hill and Häder s experiments with data set C Plots showing theoretical and simulated long-time p.d.f., f(θ), for data set C1 with τ = Plots showing theoretical and simulated long-time p.d.f., f(θ), for data set C4 with τ =

15 xiv 7.7 Plots showing the first angular moment a 1 against k 0 for the sinusoidal reorientation model, with (a) B 1 = 0.1, (b) B 1 = 0.5. Legend: theoretical results ( ), simulation results with τ = 0.1 s (- -), simulation results with τ = 1 s ( ) Plots showing the first angular moment a 1 against k 0 for the linear reorientation model, with (a) B 1 = 0.1, (b) B 1 = 0.5. Legend: theoretical results ( ), simulation results with τ = 0.1 s (- -), simulation results with τ = 1 s ( ) Plots showing the third angular moment a 3 against k 0 with B 1 = 0.5, for (a) sinusoidal reorientation model (b) linear reorientation model. Legend: theoretical results ( ), simulation results with τ = 0.1 s (- -), simulation results with τ = 1 s ( ) Plots showing the fourth angular moment a 4 against k 0 with B 1 = 0.5, for (a) sinusoidal reorientation model (b) linear reorientation model. Legend: theoretical results ( ), simulation results with τ = 0.1 s (- -), simulation results with τ = 1 s ( ) Plots showing the effect of changing the sampling length τ s of an individual random walk Plots showing how µ δ (θ) changes with θ for the sinusoidal model with various sampling lengths τ s. Simulation results for angular bins of π 9 rads ( ), and functions fitted by inspection to the data (- -) Plots showing how σδ 2 (θ) changes with θ for the sinusoidal model with various sampling lengths τ s. Simulation results for angular bins of π 9 rads ( ), and the mean from the data averaging over all θ (- -) Plots showing (a) the amplitude of the mean turning angle d τs, (b) variance of the turning angle σ 2 δ, against rescaled sampling length τ s/τ for the sinusoidal model Plots showing how µ δ (θ) changes with θ for the linear model with various sampling lengths τ s. Simulation results for angular bins of π 9 rads ( ), and functions fitted by inspection to the data (- -) Plots showing how σδ 2 (θ) changes with θ for the linear model with various sampling lengths τ s. Simulation results for angular bins of π 9 rads ( ), and the mean from the data averaging over all θ (- -) Plots showing (a) the amplitude of the mean turning angle d τs, (b) variance of the turning angle σ 2 δ, against rescaled sampling length τ s/τ for the linear model Plots showing how µ δ (θ) and σδ 2 change with θ for the sinusoidal model with fixed time between turns

16 xv 7.19 Plots showing how µ δ (θ) and σδ 2 change with θ for the linear model with fixed time between turns Plots showing (a) the amplitude of the mean turning angle d τs, (b) variance of the turning angle σ 2 δ, against rescaled sampling length τ s/τ for the sinusoidal model with fixed time between turns Plots showing (a) the amplitude of the mean turning angle d τs, (b) variance of the turning angle σ 2 δ, against rescaled sampling length τ s/τ for the linear model with fixed time between turns Log-plot of log 10 (τ) against ρ Plots showing H y (100) against κ for sinusoidal and linear reorientation for d τ = 0.1 ( ), and d τ = 0.3 ( ) Plots showing σ 2 x(100) against κ for sinusoidal and linear reorientation for d τ = 0.1 ( ), and d τ = 0.3 ( ) Plots showing σy 2 (100) against κ for sinusoidal and linear reorientation for d τ = 0.1 ( ), and d τ = 0.3 ( ) Plots showing distribution at t = 100 for sinusoidal and linear reorientation for d τ = 0.1 and κ = 0.1, κ = 10 and κ = Plots showing H y (100) against d τ for sinusoidal and linear reorientation for κ = 1 ( ), and κ = 4 ( ) Plots showing σ 2 x (100) against d τ for sinusoidal and linear reorientation for κ = 1 ( ), and κ = 4 ( ) Plots showing σ 2 y(100) against d τ for sinusoidal and linear reorientation for κ = 1 ( ), and κ = 4 ( ) Plots showing distribution at t = 100 for sinusoidal and linear reorientation for κ = 4 and d τ = 0.1, d τ = 1 and d τ = Plot of d opt against κ for sinusoidal reorientation Plot of κ(y) against y with κ I = 1, for p = 0.01 ( ), p = 0.05 ( ), and p = 0.1 (- -) Plots showing individual random walks for sinusoidal reorientation with κ(y) for various parameter values. (The scale of each plot is different) Plots showing H y (100) against p for sinusoidal and linear reorientation for d τ = 0.1 ( ), and d τ = 0.3 ( ) Plots showing σx 2 (100) against p for sinusoidal and linear reorientation for d τ = 0.1 ( ), and d τ = 0.3 ( ) Plots showing σ 2 y(100) against p for sinusoidal and linear reorientation for d τ = 0.1 ( ), and d τ = 0.3 ( ) Plots showing distribution at t = 100 for sinusoidal and linear reorientation for d τ = 0.1 and p = 0.05 and p =

17 xvi 8.16 Plot of d τ (y) against y with d int = 0.1 and d opt = 1, for q = 0.01 ( ), q = 0.05 ( ), and q = 0.1 (- -) Plots showing individual random walks for sinusoidal reorientation with d τ (y) for various parameter values. (The scale of each plot is different) Plots showing H y (100) against q for sinusoidal and linear reorientation for κ = 1 ( ), and κ = 4 ( ) Plots showing σ 2 x(100) against q for sinusoidal and linear reorientation for κ = 1 ( ), and κ = 4 ( ) Plots showing σ 2 y(100) against q for sinusoidal and linear reorientation for κ = 1 ( ), and κ = 4 ( ) Plots showing distribution at t = 100 for sinusoidal and linear reorientation for κ = 4 and q = 0.01 and q = Plots showing individual random walks for sinusoidal and linear reorientation where the preferred direction is to a point Plots showing the average position in the y-direction, H y (t), against t for sinusoidal and linear reorientation Plots showing the spread in the x-direction, σ 2 x(t), against t for sinusoidal and linear reorientation Plots showing the spread in the y-direction, σy 2 (t), against t for sinusoidal and linear reorientation Plots showing distribution at t = 100 for sinusoidal and linear reorientation where the preferred direction is to a point Plots of the spread of a population of 500 walkers after t = 100, moving as an unbiased and correlated velocity jump process with (a) κ = 1, (b) κ = 50. The dotted circle shows the maximum possible displacement at t = Plots comparing expected values of Z(c, t) ( ) to simulated results (+) for (a) κ = 1, (b) κ = 4, (c) κ = 10, (d) κ = Plots of M DD(t) v t for the velocity jump process model ( ), Kareiva & Shigesada s model ( ), Bovet & Benhamou s model (- -), and simulation results (+) Plots of MDD(t) v t for velocity jump process model with Z(c,t) ( ), Z = 0.89 (- -), and simulation results (+) Simulated plots of the spread of a population of 500 walkers after t = 100, moving as a biased and correlated velocity jump process with d τ = 0.1 and (a) sinusoidal reorientation, κ = 1, (b) sinusoidal reorientation, κ = 50, (c) linear reorientation, κ = 1, (d) linear reorientation, κ =

18 xvii 9.6 Plots of values of Z(κ,d τ,t), Z x (κ,d τ,t), and Z y (κ,d τ,t) as a function of κ at t = 1000 from numerical simulations of sinusoidal and linear reorientation with d τ = 0.1 (- -), d τ = 0.5 ( ), and d τ = 1 (- -). The solid lines ( ) correspond to Z = or Z = 0.89 respectively, the expected values if the distribution is Normal Simple infinite reef model Plots showing (a) survival probability P R (V F,γ) against death rate for (a) µ , and (b) µ Legend: deterministic model ( ), stochastic model (- -), simulation model (+) Plots of relative survival probability RSP against P R (V F,0) from theoretical ( ) and simulation (+) results Simple circular reef model Plots showing survival probability P R (d τ,κ) for sinusoidal reorientation and Model 2 against (a) κ, for d τ = 0.1 ( ), d τ = 0.3 (- -), d τ = 0.5 ( ), and d τ = 1.0 (- -); (b) d τ, for κ = 0.4 ( ), κ = 1.0 (- -), κ = 2.0 ( ), and κ = 4.0 (- -) Plots showing survival probability P R (d τ,κ) for linear reorientation and Model 2 against (a) κ, for d τ = 0.1 ( ), d τ = 0.3 (- -), d τ = 0.5 ( ), and d τ = 1.0 (- -); (b) d τ, for κ = 0.4 ( ), κ = 1.0 (- -), κ = 2.0 ( ), and κ = 4.0 (- -) Circular reef with a constant current Plots showing survival probability P R (d τ,κ) for sinusoidal reorientation and Model 3 against (a) κ, for d τ = 0.2 ( ), d τ = 0.5 (- -), d τ = 1.0 ( ), and d τ = 1.5 (- -); (b) d τ, for κ = 1.8 ( ), κ = 2.0 (- -), κ = 3.0 ( ), and κ = 5.0 (- -) Plots showing survival probability P R (d τ,κ) for linear reorientation and Model 3 against (a) κ, for d τ = 0.1 ( ), d τ = 0.3 (- -), d τ = 0.5 ( ), and d τ = 1.0 (- -); (b) d τ, for κ = 1.4 ( ), κ = 2.0 (- -), κ = 3.0 ( ), and κ = 5.0 (- -) Plots showing survival probability P R (U,κ) v U with d τ = 0.8 for (a) sinusoidal reorientation and (b) linear reorientation. Legend: κ = 1.0 ( ), κ = 2.0 (- -), κ = 3.0 ( ), and κ = 5.0 (- -)

19 List of Tables 1.1 Swimming speed and reorientation parameters estimated by Hill & Häder for the data sets C1, C3 and C Long-time numerical solutions for V(t) with linear reorientation Long-time numerical solutions for Dx1 2 (t) with linear reorientation Long-time numerical solutions for Dx2 2 (t) with linear reorientation Long-time numerical solutions for σx1 2 (t) with linear reorientation Comparing long-time numerical solutions for H(t) Comparing long-time numerical solutions for σ 2 (t) Estimated value for the amplitude of the mean turning angle µ δ (θ), and calculated mean value of σδ 2, for the sinusoidal model with rescaled sampling length τ s /τ Estimated value for the amplitude of the mean turning angle µ δ (θ), and calculated mean value of σδ 2, for the linear model with rescaled sampling length τ s /τ Estimated value for the amplitude of the mean turning angle µ δ (θ), and calculated mean value of σδ 2, for the sinusoidal model with fixed time between turns and with rescaled sampling length τ s /τ Estimated value for the amplitude of the mean turning angle µ δ (θ), and calculated mean value of σδ 2, for the linear model with fixed time between turns and with rescaled sampling length τ s /τ Values of ρ, the ratio between the expected and observed values of B 1 with the corresponding average time step between turns in the original random walk, τ xviii

20 Chapter 1 Introduction and background 1.1 General background to random walks History Brownian motion The endless irregular motion of individual pollen particles in liquid was famously studied by the English botanist Brown (1828), and such random movement has been subsequently known as Brownian motion. At the turn of the century many eminent physicists such as Einstein (1905, 1906) and Smoluchowski (1916) were drawn to the subject. During the course of research on Brownian motion, not only random walk theory (Uhlenbeck & Ornstein, 1930), but also such important fields as random processes, random noise, spectral analysis, and stochastic equations were developed The random walk Classical works on probability have been in existence for centuries so it is somewhat surprising that the first random walk problem only appeared in the literature in 1905 when the journal Nature (Vol. 72, p.294) published The problem of the random walker by Karl Pearson. The question posed was this: A man starts from a point 0 and walks l yards in a straight line: he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + δr from his starting point 0. The problem is one of considerable interest, but I have only succeeded in obtaining an integrated solution for two stretches. I think, however, that a solution ought to be found, if only in the form of a series in powers of 1/n, where n is large. Lord Rayleigh responded (Nature, Vol. 72, p.318, 1905): 1

21 CHAPTER 1: Introduction and background 2 The problem, proposed by Prof. Karl Pearson in the current number of Nature, is the same as that of the composition of n isoperiodic vibrations of unit amplitude and of phases distributed at random, considered in Philosophical Magazine, Vol. 10, p.73, 1880; Vol. 47, p.246, 1889 (Scientific Papers, I, p.491; IV, p.370). If n be very great, the probability sought is 2n 1 e r2 /n r dr. Probably methods similar to those employed in the papers referred to would avail for the development of an approximate expression applicable when n is only moderately great. In fact, Rayleigh had been studying similar problems to the random walk but under different names. The first simple models of movement using random walks are uncorrelated, meaning that each step taken is completely independent of previous steps taken and as the direction moved at each step is completely random the motion is Brownian. Such models can be shown to produce the standard diffusion equation (sometimes called the heat equation). Bias can be introduced by making the probability of moving in a certain direction greater and one can derive the drift-diffusion equation. These models have been classed as position jump processes (Othmer et al., 1988), and in general are only valid for large time scales as their solutions allow for effectively infinite propagation speeds. They can be thought of as an asymptotic approximation to the true equations governing movement that include correlation effects The isotropic random walk and the diffusion equation The simple isotropic random walk model is the basis of most of the theory of diffusive processes. The derivation of the probability distribution is a standard procedure (see for example Chandreskar (1943), Lin & Segel (1974), Okubo (1980), Murray (1993) etc.). The main points are presented here Deriving an equation for the probability density For the simple isotropic one-dimensional random walk it is straightforward to derive an equation for the probability density function by considering the limit as the number of steps gets very large. Consider a one dimensional uniform lattice, and suppose we have a walker moving along the lattice. The walker moves a short distance δ either left or right in a short time τ. The motion is assumed to be completely random (isotropic) so that the probability of moving left or right is 1 2. After one time interval, τ, the walker can either be a distance of δ to the left of the origin with probability 1 2, or a distance of δ to the

22 CHAPTER 1: Introduction and background 3 right of the origin with probability 1 2. After the next time interval, the walker will either be a distance of 2δ to the left of the origin with probability 1 4, or a distance of 2δ to the right of the origin with probability 1 4, or will have returned to the origin with probability 1 2 (but the walker cannot still be a distance δ from the origin the walker can only be an even distance from the origin). Continuing in this way, the probability that a walker will be at a distance of mδ to the right of the origin after N time steps (where m and N are even), is given by p(m,n) = ( 1 N! 2 )N [(N + m)/2]![(n m)/2]! = (1 2 )N N N m 2. (1.1) This is the binomial distribution, which for large N converges to the Gaussian (or Normal) distribution, see for example Clarke & Cooke (1992). Thus, ( )1 2 lim p(m,n) = 2 e m 2 /2N. (1.2) N πn Let x = mδ, and t = τn, and since m is even we set ( x P(x,t) dx p δ, t ) dx τ 2δ. (1.3) Then the probability of being between x and x + dx is given by P(x,t) dx = 1 2πδ 2 t/τ e x2 τ/2δ 2t dx, (1.4) and if we take limits such that τ, δ 0, while δ 2 /τ = constant 2D, then P(x,t) = 1 4πDt e x2 /4Dt. (1.5) For x R and t R +, equation (1.5) is the fundamental solution to the diffusion equation P t = D 2 P 2 x, (1.6) where P(x, 0) = δ(x) (where δ is the Dirac delta function). If we multiply equation (1.6) by N, the number of individual walkers in a population, then we get a special case (where D is constant) of Fick s equation (1.7) for the concentration (C), or number density of the population (see Okubo (1980)) C t = ( x Solution plots for (1.5) are shown in Figure 1.1. D C x ). (1.7) Useful statistics of this process are the mean position, < x >, and the mean squared displacement, < x 2 >, defined as < x >= xp(x, t) dx, (1.8)

23 CHAPTER 1: Introduction and background P(x,t) P(x,t) x D=1 D=5 D= x D=1 D=5 D=10 (a) P(x,1) (b) P(x,10) Figure 1.1: Plots showing P(x,t) for D = 1, 5 and 10, and t = 1 and 10. (The scales on each plot are different). and < x 2 >= x 2 P(x,t) dx. (1.9) For the one-dimensional diffusion solution, < x >= 0 (as we have no bias or preferred direction), and < x 2 >= 2Dt. It is a standard result for a diffusion process that the mean squared displacement increases in proportion to time, < x 2 > t Solutions in higher dimensions A similar derivation can be completed in higher dimensions. In s dimensions, the diffusion equation is given by (Montroll & Shlesinger, 1984) ( ) P 2 t = D x x 2 P. (1.10) s If we assume an initial delta function distribution P(x 1,...,x s,0) = δ(x 1 )...δ(x s ), then (1.10) has solution where r 2 = x x2 s. P(x,t) = From (1.10) the two-dimensional diffusion equation is 1 (4πDt) s/2e r2 /4Dt, (1.11) ( P 2 ) t = D P x P y 2, (1.12)

24 CHAPTER 1: Introduction and background 5 where P(x, y, 0) = δ(x)δ(y). From (1.11), the solution is P(x,y,t) = 1 4πDt e (x2 +y 2 )/4Dt. (1.13) Plots of example solutions for (1.13) are shown in Figure 1.2 for different diffusion coefficients, D P(x,y,t) 0.04 P(x,y,t) y x y x (a) P(x,y,1) for D = 1 (b) P(x,y,10) for D = P(x,y,t) P(x,y,t) y x y x (c) P(x,y,1) for D = 5 (d) P(x,y,10) for D = 5 Figure 1.2: Plots showing P(x,y,t) for D = 1 and 5, and t = 1 and 10. (The scales on each plot are different). For the two-dimensional diffusion solution, the mean position is < (x, y) >= (0, 0), and the mean squared displacement is < r 2 >=< (x 2 + y 2 ) >= 4Dt. Looking at the solutions in (1.5) and (1.13), one can see that P(x,t) > 0 for any t > 0 and any x, y R. The diffusion process predicts a non-zero probability for arbitrarily large displacements at arbitrarily small times, and in this sense the underlying speed of propagation is infinite. Because of this, and because in (1.2) and (1.5) we assumed that N and δ 0, the solution of the diffusion equation can be considered as an