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Biased random walks in biology

Profile image of Edward CodlingEdward Codling

2003

Abstract
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AI

This research explores the concept of biased random walks and their applications in biological systems, focusing on the historical development of random walk theory and its significance in understanding biological processes. Key findings demonstrate the differences in directional bias during random walks, particularly in the context of fish larvae movement and other biological phenomena. The work contributes to theoretical advancements in modeling such movements, offering insights into ecological dynamics and behavioral patterns.

Figures (170)
Figure 1.1: Plots showing P(az,t) for D = 1, 5 and 10, and t = 1 and 10. (The scales on each plot are different). A similar derivation can be completed in higher dimensions. In s dimensions, the diffusion
Figure 1.1: Plots showing P(az,t) for D = 1, 5 and 10, and t = 1 and 10. (The scales on each plot are different). A similar derivation can be completed in higher dimensions. In s dimensions, the diffusion
Figure 1.3: Plots of P(z,t), the solution of the drift diffusion equation, for various u
Figure 1.3: Plots of P(z,t), the solution of the drift diffusion equation, for various u
We can expand these equations as Taylor series to give
We can expand these equations as Taylor series to give
Figure 1.4: Plots of p(x,t), the solution of the telegraph equation for various parameter From Abramowitz and Stegun (1965), the Bessel functions have the following asymptotic expansions
Figure 1.4: Plots of p(x,t), the solution of the telegraph equation for various parameter From Abramowitz and Stegun (1965), the Bessel functions have the following asymptotic expansions
Figure 1.6: Examples of the von Mises distribution for various values of «, and yw = 0.
Figure 1.6: Examples of the von Mises distribution for various values of «, and yw = 0.
This relation between « and o? is plotted in Figure 1.7. The function A;(«) and its inverse A~*(«) are readily computed numerically using com-
This relation between « and o? is plotted in Figure 1.7. The function A;(«) and its inverse A~*(«) are readily computed numerically using com-
Suppose that the orientation coefficient takes the form
Suppose that the orientation coefficient takes the form
Although the experiments contained a lot of ‘noise’, they found that (1.87) fitted the data vell. Values of B~!, the amplitude of the orientation coefficient, were extrapolated back rom plotted graphs of the data. As there was some doubt over how to extrapolate the lata, different values for B~' were suggested. One estimate was made over all the sampling ime steps observed, while another was made using only those sampling time steps that were less than 0.4 s. It was found harder to verify (1.88) due to a lot of experimental 1oise. Using the expected long-time angular distributions, (1.100) and (1.102), and the sstimates of the parameter B~! they calculated a value of o?. Hence, for each estimate of B~ a different value for a2 was calculated. Using this estimate the rotational diffusivity, D = 02/2, can also be calculated. Data set Cl corresponds to C. nivalis moving in a vertical plane and exhibiting gyrotaxis,
Although the experiments contained a lot of ‘noise’, they found that (1.87) fitted the data vell. Values of B~!, the amplitude of the orientation coefficient, were extrapolated back rom plotted graphs of the data. As there was some doubt over how to extrapolate the lata, different values for B~' were suggested. One estimate was made over all the sampling ime steps observed, while another was made using only those sampling time steps that were less than 0.4 s. It was found harder to verify (1.88) due to a lot of experimental 1oise. Using the expected long-time angular distributions, (1.100) and (1.102), and the sstimates of the parameter B~! they calculated a value of o?. Hence, for each estimate of B~ a different value for a2 was calculated. Using this estimate the rotational diffusivity, D = 02/2, can also be calculated. Data set Cl corresponds to C. nivalis moving in a vertical plane and exhibiting gyrotaxis,
Figure 2.1: Example of a two-dimensional lattice random walk. right with probabilities independent of position, given by u, d, / and r respectively, where
Figure 2.1: Example of a two-dimensional lattice random walk. right with probabilities independent of position, given by u, d, / and r respectively, where
Figure 2.2: Example of a multi-directional random walk. If we take a forward time step then the number density, f(x,y), of individuals at position
Figure 2.2: Example of a multi-directional random walk. If we take a forward time step then the number density, f(x,y), of individuals at position
2.5.2 Solution for biased movement
2.5.2 Solution for biased movement
2.5.2.3 Solution of the Fokker—Planck equation For both the sinusoidal and linear models b; = aj2 = 0, and (2.39) simplifies to
2.5.2.3 Solution of the Fokker—Planck equation For both the sinusoidal and linear models b; = aj2 = 0, and (2.39) simplifies to
Figure 2.3: Plots showing f(x, y,t) for various parameter values at t = 10. In the simple random walk models presented previously, the walkers can have a bias it their movement which is either fixed or dependent on the spatial position, but not de pendent on the previous direction of movement, i.e. there is no correlation. As suggestec in the introduction chapter, the diffusion equation and corresponding solutions can bi considered as the long-time limiting solution of the underlying process that includes cor relation effects. The main problem with all the diffusion-type equations we have derivec is that the solutions allow for effectively infinite propagation speeds. This comes abou because of the way we take certain limits, see for example Okubo (1980). We assume thai In the simple random walk models presented previously, the walkers can have a bias in
Figure 2.3: Plots showing f(x, y,t) for various parameter values at t = 10. In the simple random walk models presented previously, the walkers can have a bias it their movement which is either fixed or dependent on the spatial position, but not de pendent on the previous direction of movement, i.e. there is no correlation. As suggestec in the introduction chapter, the diffusion equation and corresponding solutions can bi considered as the long-time limiting solution of the underlying process that includes cor relation effects. The main problem with all the diffusion-type equations we have derivec is that the solutions allow for effectively infinite propagation speeds. This comes abou because of the way we take certain limits, see for example Okubo (1980). We assume thai In the simple random walk models presented previously, the walkers can have a bias in
We can now substitute for f,J/,b and r. After dividing through by 7 and taking the limit
We can now substitute for f,J/,b and r. After dividing through by 7 and taking the limit
Figure 3.1: Sketch of the probability distributions for h(d) and k(@) as used by Othmer et al. (1988). Figure 3.1: Sketch of the probability distributions for h(6) and k(@) as used by Othmer et
Figure 3.1: Sketch of the probability distributions for h(d) and k(@) as used by Othmer et al. (1988). Figure 3.1: Sketch of the probability distributions for h(6) and k(@) as used by Othmer et
is at the origin in this case). See Figure 3.2. Figure 3.2: Sketch showing the difference between D? and o? (H is the average position).
is at the origin in this case). See Figure 3.2. Figure 3.2: Sketch showing the difference between D? and o? (H is the average position).
The following plots show the general behaviour of the solutions to the equations of Othmer 3.3.3.6 Solution plots
The following plots show the general behaviour of the solutions to the equations of Othmer 3.3.3.6 Solution plots
Figure 3.4: Plots of D?(t) and o?(t) for various values of C;.
Figure 3.4: Plots of D?(t) and o?(t) for various values of C;.
where the terms C’f; and C2 correspond to the constant terms in the revised differential equations (4.58) and (4.59) and are given by
where the terms C’f; and C2 correspond to the constant terms in the revised differential equations (4.58) and (4.59) and are given by
since a > 0 is the real part of the eigenvalues, the solutions (4.80) and (4.81) will tend to
since a > 0 is the real part of the eigenvalues, the solutions (4.80) and (4.81) will tend to
Equation (4.104) gives real-valued solutions if the eigenvalues ¢; are complex conjugates. 4.7.2.1 Solving equations for G;(t) and Go(t) Now that we have closed the original system of equations up to leading order it is straight
Equation (4.104) gives real-valued solutions if the eigenvalues ¢; are complex conjugates. 4.7.2.1 Solving equations for G;(t) and Go(t) Now that we have closed the original system of equations up to leading order it is straight
Figure 4.3: Plot of D*(t) for d; = 0.1 and d, = 0.3 and various values of «. (The scale of
Figure 4.3: Plot of D*(t) for d; = 0.1 and d, = 0.3 and various values of «. (The scale of
Figure 4.4: Plot of o?(t) for d; = 0.1 and d, = 0.3 and various values of «. (The scale of each plot is different)
Figure 4.4: Plot of o?(t) for d; = 0.1 and d, = 0.3 and various values of «. (The scale of each plot is different)
Figure 4.5: Plot of D?,(t) and D?,(t) for d; = 0.3 and various values of k. (The scale of each plot is different)
Figure 4.5: Plot of D?,(t) and D?,(t) for d; = 0.3 and various values of k. (The scale of each plot is different)
Figure 4.6: Plot of o%,(t) and o%,(t) for d, = 0.3 and various values of «. (The scale of wexclhe <eTeck See off Pew ¥
Figure 4.6: Plot of o%,(t) and o%,(t) for d, = 0.3 and various values of «. (The scale of wexclhe <eTeck See off Pew ¥
We now use the following rescalings
We now use the following rescalings
Figure 4.7: Plot of ¢; against « for d- = 0,0.1, 0.2, 0.3. As « gets larger our assumption that we can approximate the higher order moments suck
Figure 4.7: Plot of ¢; against « for d- = 0,0.1, 0.2, 0.3. As « gets larger our assumption that we can approximate the higher order moments suck
Figure 4.8: Plot of C2 against « for d- = 0.1, 0.2, 0.3. becomes negative as « increases for d; = 0.1, 0.2, 0.3.
Figure 4.8: Plot of C2 against « for d- = 0.1, 0.2, 0.3. becomes negative as « increases for d; = 0.1, 0.2, 0.3.
All other statistics we are interested in are assumed to be 0 at t = 0.
All other statistics we are interested in are assumed to be 0 at t = 0.
For simplicity, we can write the first system as The spatial higher moments can themselves be split into a system arising from the even 5.3.4 System of equations for spatial moments
For simplicity, we can write the first system as The spatial higher moments can themselves be split into a system arising from the even 5.3.4 System of equations for spatial moments
Similarly, we can write the second system as where the dj; correspond to the constant coefficients (dependent on d, and « only) from
Similarly, we can write the second system as where the dj; correspond to the constant coefficients (dependent on d, and « only) from
Table 5.1: Long-time numerical solutions for V(t) with linear reorientation We have shown in (5.94) that the long-time solution for H(t) is given by We have shown in (5.92) that the long-time solution for V(t) is given by
Table 5.1: Long-time numerical solutions for V(t) with linear reorientation We have shown in (5.94) that the long-time solution for H(t) is given by We have shown in (5.92) that the long-time solution for V(t) is given by
Table 5.2: Long-time numerical solutions for D?,(t) with linear reorientation
Table 5.2: Long-time numerical solutions for D?,(t) with linear reorientation
Table 5.3: Long-time numerical solutions for D?5(t) with linear reorientation
Table 5.3: Long-time numerical solutions for D?5(t) with linear reorientation
Table 5.4: Long-time numerical solutions for 2, (t) with linear reorientation
Table 5.4: Long-time numerical solutions for 2, (t) with linear reorientation
Figure 5.10: Plot of H(t) for d, = 0.1 and d, = 0.3 and various values of kK. Figure 5.9: Plot of V(t) for d, = 0.1 and d, = 0.3 and various values of &.
Figure 5.10: Plot of H(t) for d, = 0.1 and d, = 0.3 and various values of kK. Figure 5.9: Plot of V(t) for d, = 0.1 and d, = 0.3 and various values of &.
the two statistics.
the two statistics.
dimensionalised long-time general solutions, (4.95) etc, derived in the previous chapter. is what is suggested by the results presented in Table (5.5).
dimensionalised long-time general solutions, (4.95) etc, derived in the previous chapter. is what is suggested by the results presented in Table (5.5).
5.6.2 Comparing solutions for o7(t) Looking at the results in Table (5.6) one can see that the long-time spread about the mean
5.6.2 Comparing solutions for o7(t) Looking at the results in Table (5.6) one can see that the long-time spread about the mean
Figure 6.1: Simple algorithm for an individual random walk. 5.2.1 Simulation of an individual random walk 5.2 Computer simulations of random walks
Figure 6.1: Simple algorithm for an individual random walk. 5.2.1 Simulation of an individual random walk 5.2 Computer simulations of random walks
Figure 6.2 shows that when « ~ 0 and d; = 0, the motion is close to random (Browniar notion). Figure 6.3 shows that when « is large but d, = 0, the motion is more correlated
Figure 6.2 shows that when « ~ 0 and d; = 0, the motion is close to random (Browniar notion). Figure 6.3 shows that when « is large but d, = 0, the motion is more correlated
correlated but there is no overall preferred direction.
correlated but there is no overall preferred direction.
Figure 6.4: iii) Random walk with « = 0.5, d, = 0.2. The random walk is less correlated
Figure 6.4: iii) Random walk with « = 0.5, d, = 0.2. The random walk is less correlated
Figure 6.8: Plots showing theoretical H,(t) (—), and 95% confidence interval from sim-
Figure 6.8: Plots showing theoretical H,(t) (—), and 95% confidence interval from sim-
Figure 6.9: Plots showing theoretical H,(t) (—), and 95% confidence interval from sim-
Figure 6.9: Plots showing theoretical H,(t) (—), and 95% confidence interval from sim-
Figure 6.20: Plots showing theoretical (—), and simulated (---), D?(t) against time for
Figure 6.20: Plots showing theoretical (—), and simulated (---), D?(t) against time for
Figure 6.24: Plots showing theoretical (—), and simulated (---), o?(t) against time for
Figure 6.24: Plots showing theoretical (—), and simulated (---), o?(t) against time for
Figure 6.25: Plots showing theoretical (—), and simulated (---), o?(t) against time for
Figure 6.25: Plots showing theoretical (—), and simulated (---), o?(t) against time for
Figure 6.27: Plots showing theoretical (—), and simulated (---), o?(t) against time for
Figure 6.27: Plots showing theoretical (—), and simulated (---), o?(t) against time for
Figure 6.33: Plots showing o; (t) against time for linear reorientation with various values
Figure 6.33: Plots showing o; (t) against time for linear reorientation with various values
Figure 6.34: Example plots of the population position and spread at t = 100
Figure 6.34: Example plots of the population position and spread at t = 100
Figure 6.36: Plots showing D?(100) against « for sinusoidal and linear reorientation with or d,; = 0,0.1,0.2 and 0.3. Figure 6.36 shows a good match between the theoretical and 6.4.2.2 Spread about the origin D?(t)
Figure 6.36: Plots showing D?(100) against « for sinusoidal and linear reorientation with or d,; = 0,0.1,0.2 and 0.3. Figure 6.36 shows a good match between the theoretical and 6.4.2.2 Spread about the origin D?(t)
increases from 0 to 8, for d-; = 0,0.1,0.2 and 0.3. Figure 6.37 shows a good matcl simulated solutions. From the plots the general behaviour as « increases seems to be a slow maximum value and then decreasing towards zero. For a particular value of « the spread is
increases from 0 to 8, for d-; = 0,0.1,0.2 and 0.3. Figure 6.37 shows a good matcl simulated solutions. From the plots the general behaviour as « increases seems to be a slow maximum value and then decreasing towards zero. For a particular value of « the spread is
Simulations using the linear reorientation model with these reorientation parameters and ures 6.45(b) — 6.45(d) show a good match between theoretical and simulated results.
Simulations using the linear reorientation model with these reorientation parameters and ures 6.45(b) — 6.45(d) show a good match between theoretical and simulated results.
When closing the systems of differential equations for the linear reorientation model in Chapter 5, we approximated the higher order moments F3(t), K3(t), M3(t) and also the moment Mo(t) as time-independent constants using the steady state solution (7.6). Using simulations we can demonstrate that the approximation for Mo(t) as a time-independent constant is reasonable. Figure 7.1 compares values of Mo(t) for simulated data and ap- proximations using the steady state solution (7.6). Note that assuming there is a uniform spread of initial directions, Mo(0) = 77/3 ~ 3.29. The simulations have been completed with three values of & and all have d, = 0.2 and A = 1. It is clear from Figure 7.1 that with three values of « and all have d, = 0.2 and A = 1. It is clear from Figure 7.1 that he simulated values reach a steady state very quickly. Thus, although the approxim ons using Hill & Hader’s steady state solution, (7.6), seem to slightly underestimate t mulated values, it seems reasonable to approximate Mpo(t) as a time-independent co fant as we have done in Chapter 5. The reason for this slight underestimation of t. mulation results is discussed later in this chapter and seems likely to be because Hill fader’s steady state approximation, (7.6), is valid for continuous random walks. If 1 se a smaller time step between turns (larger value of \) then the approximations becor loser to the simulated results, although the simulations now take longer to reach t. teady state solutions. the simulated values reach a steady state very quickly. Thus, although the approxima-
When closing the systems of differential equations for the linear reorientation model in Chapter 5, we approximated the higher order moments F3(t), K3(t), M3(t) and also the moment Mo(t) as time-independent constants using the steady state solution (7.6). Using simulations we can demonstrate that the approximation for Mo(t) as a time-independent constant is reasonable. Figure 7.1 compares values of Mo(t) for simulated data and ap- proximations using the steady state solution (7.6). Note that assuming there is a uniform spread of initial directions, Mo(0) = 77/3 ~ 3.29. The simulations have been completed with three values of & and all have d, = 0.2 and A = 1. It is clear from Figure 7.1 that with three values of « and all have d, = 0.2 and A = 1. It is clear from Figure 7.1 that he simulated values reach a steady state very quickly. Thus, although the approxim ons using Hill & Hader’s steady state solution, (7.6), seem to slightly underestimate t mulated values, it seems reasonable to approximate Mpo(t) as a time-independent co fant as we have done in Chapter 5. The reason for this slight underestimation of t. mulation results is discussed later in this chapter and seems likely to be because Hill fader’s steady state approximation, (7.6), is valid for continuous random walks. If 1 se a smaller time step between turns (larger value of \) then the approximations becor loser to the simulated results, although the simulations now take longer to reach t. teady state solutions. the simulated values reach a steady state very quickly. Thus, although the approxima-
Both sets of parameter values produce the same long-time theoretical probability distr oution for f(A). This is expected since Hill & Hader used the observed long-time angula listribution from experiments to estimate of given two different estimates of B~! (Sectio 1.5.3). Even with 2000 data points our simulation data is fairly noisy, but the simulate results seem a reasonable fit to the theoretical distribution. It would be possible to d various tests to see how good the fit between theory and simulation is (see for exampk Fisher (1993), or Mardia & Jupp (1999)), but for our purposes, it is easier to compare b looking at the moments of the distributions as we do in the next section. Plots of the long-time angular distribution using parameters from Hill & Hader’s experi
Both sets of parameter values produce the same long-time theoretical probability distr oution for f(A). This is expected since Hill & Hader used the observed long-time angula listribution from experiments to estimate of given two different estimates of B~! (Sectio 1.5.3). Even with 2000 data points our simulation data is fairly noisy, but the simulate results seem a reasonable fit to the theoretical distribution. It would be possible to d various tests to see how good the fit between theory and simulation is (see for exampk Fisher (1993), or Mardia & Jupp (1999)), but for our purposes, it is easier to compare b looking at the moments of the distributions as we do in the next section. Plots of the long-time angular distribution using parameters from Hill & Hader’s experi
sampling time steps for T, < 0.4 s (see Section 1.5.3). in the next section.
sampling time steps for T, < 0.4 s (see Section 1.5.3). in the next section.
Figure 7.4: Plots showing theoretical and simulated long-time p.d.f., f(@), with parameter the same theoretical distribution for all values of r. However when deriving the Fokker-
Figure 7.4: Plots showing theoretical and simulated long-time p.d.f., f(@), with parameter the same theoretical distribution for all values of r. However when deriving the Fokker-
Figure 7.5: Plots showing theoretical and simulated long-time p.d.f., f(@), for data set C1
Figure 7.5: Plots showing theoretical and simulated long-time p.d.f., f(@), for data set C1
Figure 7.6: Plots showing theoretical and simulated long-time p.d.f., f(@), for data set C4
Figure 7.6: Plots showing theoretical and simulated long-time p.d.f., f(@), for data set C4
of 7 = 0.1 s, and the simulated result using a time step of T= 1s. 7.2.3.2 Third angular moment, n = 3 amount of relative noise in the simulation data. For this reason results are not shown.
of 7 = 0.1 s, and the simulated result using a time step of T= 1s. 7.2.3.2 Third angular moment, n = 3 amount of relative noise in the simulation data. For this reason results are not shown.
using a time step of 7 = 0.1 s, and the simulated result using a time step of T= 1 s. The plots in Figure 7.10 show how the fourth angular moment a4 changes as the con-
using a time step of 7 = 0.1 s, and the simulated result using a time step of T= 1 s. The plots in Figure 7.10 show how the fourth angular moment a4 changes as the con-
Plots showing o7(0) against @ for various sampling lengths 7, are shown in Figure 7.13. In the above plots the straight line plotted is the mean of the variance, o7(0), averaged as there seems to be little noise in the data (except for 7, = 20).
Plots showing o7(0) against @ for various sampling lengths 7, are shown in Figure 7.13. In the above plots the straight line plotted is the mean of the variance, o7(0), averaged as there seems to be little noise in the data (except for 7, = 20).
Table 7.1: Estimated value for the amplitude of the mean turning angle ju5(@), and calcu- lated mean value of 03, for the sinusoidal model with rescaled sampling length 7,/T. 7.3.2.3 Estimating the reorientation parameters
Table 7.1: Estimated value for the amplitude of the mean turning angle ju5(@), and calcu- lated mean value of 03, for the sinusoidal model with rescaled sampling length 7,/T. 7.3.2.3 Estimating the reorientation parameters
from the gradient of the observed values (and recalling that 7 = 0.1) is given by
from the gradient of the observed values (and recalling that 7 = 0.1) is given by
From Figure 7.17 we have very similar behaviour to the results observed from the sinusoidal
From Figure 7.17 we have very similar behaviour to the results observed from the sinusoidal
Figure 7.18: Plots showing how j15(@) and o% change with @ for the sinusoidal model with
Figure 7.18: Plots showing how j15(@) and o% change with @ for the sinusoidal model with
The data in Table 7.5 has been plotted as a log-plot (with base 10) in Figure 7.22. The results that are plotted in Figure 7.22 confirm the suggestion that as our velocity jump
The data in Table 7.5 has been plotted as a log-plot (with base 10) in Figure 7.22. The results that are plotted in Figure 7.22 confirm the suggestion that as our velocity jump
less random and more like a straight line. If we assume that the ‘optimal’ motion is that which produces the largest displacement in 3.2.1 The effect of the parameter «
less random and more like a straight line. If we assume that the ‘optimal’ motion is that which produces the largest displacement in 3.2.1 The effect of the parameter «
Figure 8.5: Plots showing H,(100) against d, for sinusoidal and linear reorientation for d- further then there is a decrease in the average displacement. the maximum average displacement appears to be when d, = 1.
Figure 8.5: Plots showing H,(100) against d, for sinusoidal and linear reorientation for d- further then there is a decrease in the average displacement. the maximum average displacement appears to be when d, = 1.
Figure 8.6: Plots showing o2(100) against d, for sinusoidal and linear reorientation for
Figure 8.6: Plots showing o2(100) against d, for sinusoidal and linear reorientation for
Figure 8.7: Plots showing a, (100) against d, for sinusoidal and linear reorientation for
Figure 8.7: Plots showing a, (100) against d, for sinusoidal and linear reorientation for
changes as « increases. It is clear from Figure 8.9 that when « * 0, and the angula: Figure 8.9: Plot of dop: against « for sinusoidal reorientation. distribution is almost uniform, then doy, ~ 5 as calculated previously. As k increases, zt 2
changes as « increases. It is clear from Figure 8.9 that when « * 0, and the angula: Figure 8.9: Plot of dop: against « for sinusoidal reorientation. distribution is almost uniform, then doy, ~ 5 as calculated previously. As k increases, zt 2
parameter values: p = 0.01, 0.05 and 0.1. Our original theoretical model is a special case of this general model, where p = 0. 8.3.1.2 Examples of individual random walks The plots in Figure 8.11 show examples of simulations of individual random walks using
parameter values: p = 0.01, 0.05 and 0.1. Our original theoretical model is a special case of this general model, where p = 0. 8.3.1.2 Examples of individual random walks The plots in Figure 8.11 show examples of simulations of individual random walks using
as constants in all simulations. Each simulation is run for t = 0 up to t = 1000. s directed back to the preferred direction and once moving in this direction, is unlikely
as constants in all simulations. Each simulation is run for t = 0 up to t = 1000. s directed back to the preferred direction and once moving in this direction, is unlikely
parameter p increases from 0 to 1, for d; = 0.1 and d, = 0.3. The plots in Figure 8.12 show how the simulated mean position at t = 100, H,(100), fo: Figure 8.12 shows similar behaviour to Figure 8.1 in that the average displacement in the preferred direction increases asymptotically towards H,,(100) = 100 as the parameter 4 increases, with the greatest rate of increase for the smaller values of p. However, unlike Figure 8.1, if p = 0 then H, 4 0 — this is simply because p measures how the parameter kK changes with the y-position. If p = 0 then we simply have our previous spatially independent model where « is fixed for all (x,y). As observed in previous simulation, the greatest displacement is for the larger values of d,, and the linear model produces a larger displacement than the sinusoidal model for the same parameter values.
parameter p increases from 0 to 1, for d; = 0.1 and d, = 0.3. The plots in Figure 8.12 show how the simulated mean position at t = 100, H,(100), fo: Figure 8.12 shows similar behaviour to Figure 8.1 in that the average displacement in the preferred direction increases asymptotically towards H,,(100) = 100 as the parameter 4 increases, with the greatest rate of increase for the smaller values of p. However, unlike Figure 8.1, if p = 0 then H, 4 0 — this is simply because p measures how the parameter kK changes with the y-position. If p = 0 then we simply have our previous spatially independent model where « is fixed for all (x,y). As observed in previous simulation, the greatest displacement is for the larger values of d,, and the linear model produces a larger displacement than the sinusoidal model for the same parameter values.
Figure 8.18: Plots showing H,(100) against q for sinusoidal and linear reorientation for parameter q increases from 0 to 1, for « = 1 and & = 4. The plots in Figure 8.18 show that as the parameter g increases there is a slow gradual increase in the average displacement in the preferred direction. In general, the linear model produces the larger displacement, and there is also larger displacement for the larger values of k. When g = 0 there is a fixed (non-zero) value of d, and hence H,,(100) is also non-zero. For larger values of gq the plots seem to tend to an asymptotic limit that is not the maximum allowed displacement. The parameter q controls how quickly the walkers reach the optimal value of d- = dop¢ as they move in the positive y-direction. However, even if d; = dop¢ for all (x,y) the average displacement is still dependent on the value of «, see Figure 8.5, and this & value will control what this asymptotic limit will
Figure 8.18: Plots showing H,(100) against q for sinusoidal and linear reorientation for parameter q increases from 0 to 1, for « = 1 and & = 4. The plots in Figure 8.18 show that as the parameter g increases there is a slow gradual increase in the average displacement in the preferred direction. In general, the linear model produces the larger displacement, and there is also larger displacement for the larger values of k. When g = 0 there is a fixed (non-zero) value of d, and hence H,,(100) is also non-zero. For larger values of gq the plots seem to tend to an asymptotic limit that is not the maximum allowed displacement. The parameter q controls how quickly the walkers reach the optimal value of d- = dop¢ as they move in the positive y-direction. However, even if d; = dop¢ for all (x,y) the average displacement is still dependent on the value of «, see Figure 8.5, and this & value will control what this asymptotic limit will
Figure 8.19: Plots showing o2(100) against qg for sinusoidal and linear reorientation for
Figure 8.19: Plots showing o2(100) against qg for sinusoidal and linear reorientation for
Figure 8.20: Plots showing o2(100) against q for sinusoidal and linear reorientation for
Figure 8.20: Plots showing o2(100) against q for sinusoidal and linear reorientation for
The plots in Figure 8.22 show examples of simulations of individual random walks using towards the point source and then stay in the vicinity of it rather than moving in the 8.4.2 Examples of individual random walks
The plots in Figure 8.22 show examples of simulations of individual random walks using towards the point source and then stay in the vicinity of it rather than moving in the 8.4.2 Examples of individual random walks
Figure 8.26: Plots showing distribution at t = 100 for sinusoidal and linear reorientatior
Figure 8.26: Plots showing distribution at t = 100 for sinusoidal and linear reorientatior
Figure 9.1: Plots of the spread of a population of 500 walkers after t = 100, moving as an unbiased and correlated velocity jump process with (a) « = 1, (b) k = 50. The dotted circle shows the maximum possible displacement at t = 100. quickly. This predicted behaviour can be clearly seen in Figure 9.1, for correlated random
Figure 9.1: Plots of the spread of a population of 500 walkers after t = 100, moving as an unbiased and correlated velocity jump process with (a) « = 1, (b) k = 50. The dotted circle shows the maximum possible displacement at t = 100. quickly. This predicted behaviour can be clearly seen in Figure 9.1, for correlated random
the results expected from the other two equations. This is because of the point made in Section 9.2.2 — the equation of Bovet & Benhamou does not take into account the variable step length that is present. From Figure 9.3(b), for « = 20 the equation of Bovet & Benhamou overestimates both the simulated results and the results expected from the other two equations. This is because the equation of Bovet & Benhamou is only valid as the long-time limiting solution when the spatial distribution has settled down and is Normal. A highly correlated random walk (K = 20) will take a longer time before Bovet & Benhamou’s equation is valid. If we look at a longer time period (not shown) the equation of Bovet & Benhamou starts to underestimate the simulated results because of the point made above about the variable step length. the results expected from the other two equations. This is because of the point made
the results expected from the other two equations. This is because of the point made in Section 9.2.2 — the equation of Bovet & Benhamou does not take into account the variable step length that is present. From Figure 9.3(b), for « = 20 the equation of Bovet & Benhamou overestimates both the simulated results and the results expected from the other two equations. This is because the equation of Bovet & Benhamou is only valid as the long-time limiting solution when the spatial distribution has settled down and is Normal. A highly correlated random walk (K = 20) will take a longer time before Bovet & Benhamou’s equation is valid. If we look at a longer time period (not shown) the equation of Bovet & Benhamou starts to underestimate the simulated results because of the point made above about the variable step length. the results expected from the other two equations. This is because of the point made
Figure 10.1: Simple ‘infinite’ reef model. 10.3.2 Deterministic model for population dynamics
Figure 10.1: Simple ‘infinite’ reef model. 10.3.2 Deterministic model for population dynamics
Simulations were completed with 100,000 walkers all moving with the random walk mode described in Section 10.3.1, and subject to a Poisson process death rate with values of the parameter jz between 0.0001 s~! and 0.0004 s~!. Figure 10.2(a) and (b) show the surviva probability against y for the deterministic model (10.3), the stochastic model (10.6) anc the simulation results Pr(sim), with (a) 0.0001 < w < 0.0002, and (b) 0.0002 < yw < 0.0004. We have displayed two separate plots to highlight the relative difference betweer the deterministic, stochastic and simulated results at very small survival probabilities. he deterministic, stochastic and simulated results at very small survival probabilities. 0.0004. We have displayed two separate plots to highlight the relative difference between
Simulations were completed with 100,000 walkers all moving with the random walk mode described in Section 10.3.1, and subject to a Poisson process death rate with values of the parameter jz between 0.0001 s~! and 0.0004 s~!. Figure 10.2(a) and (b) show the surviva probability against y for the deterministic model (10.3), the stochastic model (10.6) anc the simulation results Pr(sim), with (a) 0.0001 < w < 0.0002, and (b) 0.0002 < yw < 0.0004. We have displayed two separate plots to highlight the relative difference betweer the deterministic, stochastic and simulated results at very small survival probabilities. he deterministic, stochastic and simulated results at very small survival probabilities. 0.0004. We have displayed two separate plots to highlight the relative difference between
Che simulations were completed using the sinusoidal reorientation model — results usins she linear reorientation model are qualitatively similar, although the linear model gives < iigher survival probability with the same values for the reorientation parameters (recal shat the linear model always gives a higher absolute displacement for the same paramete! values). Similarly, if reorientation parameters that produce a larger absolute displacement we used then the survival probability is larger and the relative survival probability i: smaller, but the qualitative results are similar. Similar results might also be expected i she death rate is made smaller but the distance to the reef is made larger.
Che simulations were completed using the sinusoidal reorientation model — results usins she linear reorientation model are qualitatively similar, although the linear model gives < iigher survival probability with the same values for the reorientation parameters (recal shat the linear model always gives a higher absolute displacement for the same paramete! values). Similarly, if reorientation parameters that produce a larger absolute displacement we used then the survival probability is larger and the relative survival probability i: smaller, but the qualitative results are similar. Similar results might also be expected i she death rate is made smaller but the distance to the reef is made larger.
Figure 10.4: Simple circular reef model. With these parameters, the theoretical minimum time to reach the reef is given by tin = 4900 s, corresponding to moving directly from the origin to the edge of the reef in a straight line (i.e. Ve = s = 0.1). Such models that do not include random fluctuations of the individual movement have been considered by Armsworth (2000, 2001). The probability of reaching the reef and surviving in this extreme case is Pa(Vr = 0.1) = 0.9100. This gives an expected upper bound for the survival probabilities of the subsequent simulations that do include the effects of random reorientations in the individual movement. With these parameters, the theoretical minimum time to reach the reef is given by timjn =
Figure 10.4: Simple circular reef model. With these parameters, the theoretical minimum time to reach the reef is given by tin = 4900 s, corresponding to moving directly from the origin to the edge of the reef in a straight line (i.e. Ve = s = 0.1). Such models that do not include random fluctuations of the individual movement have been considered by Armsworth (2000, 2001). The probability of reaching the reef and surviving in this extreme case is Pa(Vr = 0.1) = 0.9100. This gives an expected upper bound for the survival probabilities of the subsequent simulations that do include the effects of random reorientations in the individual movement. With these parameters, the theoretical minimum time to reach the reef is given by timjn =
Figure 10.7: Circular reef with a constant current. interest to look at a current that produces a drift directly towards the reef relative to the
Figure 10.7: Circular reef with a constant current. interest to look at a current that produces a drift directly towards the reef relative to the
Figure 10.8: Plots showing survival probability Pr(d,,«) for sinusoidal reorientation and
Figure 10.8: Plots showing survival probability Pr(d,,«) for sinusoidal reorientation and

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