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Part I Theoretical Foundations |
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1 Elements of Probability Theory |
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3 | (20) |
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1.1 Random Variables and Probability |
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3 | (3) |
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1.2 Moments and the Characteristic Function |
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6 | (2) |
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1.3 Well-Known Probability Distributions |
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8 | (6) |
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1.3.1 Normal Distribution |
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8 | (1) |
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1.3.2 Exponential Distribution |
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9 | (1) |
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1.3.3 Uniform Distribution |
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10 | (1) |
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1.3.4 Cauchy Distribution |
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11 | (1) |
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11 | (1) |
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1.3.6 Dirac Delta Distribution |
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12 | (1) |
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1.3.7 Poisson Distribution |
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12 | (1) |
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1.3.8 Binomial Distribution |
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13 | (1) |
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1.4 Multivariate Distributions |
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14 | (3) |
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1.4.1 Conditional Probabilities |
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15 | (1) |
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1.4.2 Correlation and Covariance |
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16 | (1) |
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1.5 The Central Limit Theorem |
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17 | (1) |
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18 | (5) |
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21 | (2) |
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2 Introduction to Stochastic Processes |
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23 | (40) |
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2.1 Fluctuations and Non-determinism |
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23 | (1) |
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24 | (4) |
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28 | (2) |
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30 | (5) |
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31 | (1) |
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32 | (2) |
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34 | (1) |
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2.5 Microscopic Description of Stochastic Processes |
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35 | (5) |
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2.5.1 Stochastic Differential Equations |
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35 | (5) |
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2.6 Mesoscopic Description of Stochastic Processes |
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40 | (3) |
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2.6.1 Chapman-Kolmogorov Equation |
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40 | (3) |
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2.7 Macroscopic Description of Stochastic Processes |
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43 | (13) |
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2.7.1 The Master Equation |
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43 | (5) |
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2.7.2 The Fokker-Planck Equation |
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48 | (8) |
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2.8 Summary: Micro, Meso and Macroscopic Descriptions of a Stochastic Process |
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56 | (7) |
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59 | (4) |
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Part II Stochastic Modelling for Dispersal and Movement |
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3 Microscopic, Mesoscopic and Macroscopic Descriptions of Dispersal |
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63 | (50) |
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3.1 The Diffusion Equation |
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65 | (13) |
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3.1.1 Macroscopic Derivation |
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66 | (4) |
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3.1.2 Mesoscopic Derivation |
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70 | (2) |
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3.1.3 Microscopic Derivation: Langevin's Approach |
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72 | (3) |
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3.1.4 Fundamental Solution and Statistics |
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75 | (2) |
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3.1.5 Pathologies of the Diffusion Equation |
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77 | (1) |
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78 | (9) |
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3.2.1 The Telegrapher's Equation |
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78 | (6) |
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3.2.2 Langevin Approach to Persistent Motion |
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84 | (3) |
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87 | (7) |
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3.3.1 Combination of Diffusion with Pauses |
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88 | (4) |
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3.3.2 Combination of Diffusion with Ballistic Displacements * |
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92 | (2) |
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3.4 Externally-Directed Movement |
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94 | (8) |
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94 | (2) |
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96 | (6) |
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3.5 Dispersal in Two and Three Dimensions |
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102 | (11) |
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3.5.1 Diffusion Equations in Two and Three Dimensions |
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103 | (2) |
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3.5.2 Correlated Dispersal and Turn Angle Distributions |
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105 | (4) |
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109 | (4) |
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4 Anomalous Diffusion and Continuous-Time Random Walks |
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113 | (36) |
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4.1 What Does Anomalous Mean? |
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113 | (2) |
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4.2 General Mechanisms of Anomalous Diffusion |
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115 | (6) |
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4.2.1 Long-Range Correlations |
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116 | (2) |
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4.2.2 Non-identical Displacements |
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118 | (1) |
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4.2.3 Displacements with Non-finite Mean or Variance |
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119 | (2) |
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4.3 Diffusion on Fractals |
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121 | (3) |
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4.4 Levy Flights and Levy Walks |
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124 | (3) |
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4.5 Continuous-Time Random Walks |
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127 | (14) |
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4.5.1 Random Jump Lengths: Position of a Random Walk |
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127 | (2) |
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4.5.2 Random Waiting Times |
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129 | (2) |
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4.5.3 Formulation of Continuous-Time Random Walks |
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131 | (4) |
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4.5.4 CTRWs and Anomalous Diffusion * |
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135 | (2) |
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4.5.5 Macroscopic Limit * |
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137 | (2) |
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4.5.6 Large Waiting Times and Subdiffusion * |
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139 | (1) |
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4.5.7 Large Distance Jumps and Superdiffusion * |
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140 | (1) |
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4.6 Random Velocity Models |
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141 | (8) |
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4.6.1 Velocity Models and Anomalous Diffusion * |
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144 | (2) |
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146 | (3) |
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5 Reaction-Dispersal Models and Front Propagation |
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149 | (28) |
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150 | (2) |
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5.2 Reaction-Telegrapher's Equation |
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152 | (1) |
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5.3 Reaction-Correlated Random Walks |
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153 | (2) |
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5.4 Reaction-Dispersal Models |
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155 | (5) |
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5.4.1 Discrete-Time Models |
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155 | (1) |
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5.4.2 Continuous-Time Models * |
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156 | (2) |
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5.4.3 Including Life Statistics |
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158 | (2) |
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160 | (17) |
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5.5.1 Determination of the Front Speed |
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162 | (1) |
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5.5.2 Reaction-Diffusion Fronts |
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163 | (4) |
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5.5.3 Reaction-Dispersal Fronts * |
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167 | (8) |
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175 | (2) |
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6 Random Search Strategies |
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177 | (32) |
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6.1 Mean First-Passage Time |
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180 | (10) |
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180 | (3) |
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183 | (3) |
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6.1.3 Intermittent Searches |
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186 | (4) |
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6.2 Selective Target Detection |
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190 | (4) |
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6.3 The Extensive-Intensive Search Tradeoff |
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194 | (15) |
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203 | (6) |
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Part III Selected Applications |
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209 | (36) |
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7.1 Key Physical Factors of Cell Motion |
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210 | (3) |
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210 | (1) |
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7.1.2 Substrate and Cell Density |
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211 | (1) |
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212 | (1) |
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213 | (1) |
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7.2 Analysis of Individual Cell Trajectories |
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213 | (9) |
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7.2.1 Velocity Distributions |
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214 | (2) |
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7.2.2 Velocity Correlations |
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216 | (1) |
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7.2.3 Mean Square Displacement |
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216 | (2) |
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218 | (1) |
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7.2.5 Turn Angle Distributions |
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219 | (1) |
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220 | (1) |
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221 | (1) |
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7.3 Microscopic Descriptions of Cell Motility |
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222 | (7) |
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7.3.1 The OU Process and Its Extensions |
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222 | (4) |
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7.3.2 Passing to the Cell's Frame of Reference * |
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226 | (3) |
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7.4 Mesoscopic Descriptions of Cell Motility |
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229 | (8) |
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7.4.1 Run and Tumble with Turn Angle Distributions * |
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230 | (2) |
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7.4.2 The Velocity Jump Model * |
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232 | (2) |
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7.4.3 Two-Dimensional Random Velocity Models * |
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234 | (3) |
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7.5 Summary: Cell Motion and Superdiffusion |
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237 | (8) |
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241 | (4) |
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245 | (22) |
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8.1 Estimating Dispersal Kernels and Diffusion Coefficients from Data |
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246 | (6) |
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8.1.1 Non-parametric Estimator from 1D Dispersal Data |
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248 | (1) |
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8.1.2 Non-parametric Estimator from 1D Density Data |
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249 | (1) |
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8.1.3 Non-parametric Estimator from 2D Dispersal Data |
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250 | (1) |
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8.1.4 Non-parametric Estimator from 2D Density Data |
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250 | (1) |
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8.1.5 Estimation of the Diffusion Coefficient |
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251 | (1) |
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8.2 Estimating the Waiting-Time PDF and the Intrinsic Growth Rate from Data |
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252 | (1) |
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8.2.1 Non-parametric Estimator from Life-Tables |
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252 | (1) |
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253 | (8) |
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8.3.1 House-Finch Invasion |
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253 | (2) |
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255 | (2) |
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257 | (2) |
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8.3.4 Grey Squirrel Invasion |
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259 | (2) |
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8.4 Age-Structured Models and Plant Invasions |
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261 | (6) |
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265 | (2) |
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9 Biological Searches and Random Animal Motility |
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267 | (22) |
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9.1 Experiments in the Field |
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269 | (9) |
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9.1.1 Levy Patterns in Marine Predators: Knowing When but Not Why |
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269 | (1) |
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9.1.2 Displaced Honey Bees: Where Is Home? |
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270 | (1) |
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9.1.3 Seabirds and Fishery Discards |
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271 | (4) |
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9.1.4 Human Random Searches in a Soccer Field |
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275 | (3) |
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9.2 Experiments in the Lab |
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278 | (11) |
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9.2.1 Ants Search Strategies in Interrumpted Tandem Runs |
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278 | (1) |
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9.2.2 Desert Locusts: Run, Pause, and Tumble |
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279 | (4) |
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9.2.3 Pattern Formation in Mussels |
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283 | (1) |
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9.2.4 Cell Searching Without External Signals |
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284 | (1) |
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285 | (4) |
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A Mathematical and Numerical Tools |
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289 | (16) |
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289 | (2) |
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291 | (1) |
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292 | (1) |
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293 | (2) |
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A.5 Numerical Implementation of Langevin Equations |
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295 | (1) |
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A.6 Method of Characteristics |
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296 | (2) |
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A.7 Determination of the Moments for a 2D Model with Random Turn Angles |
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298 | (3) |
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301 | (4) |
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303 | (2) |
Index |
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305 | |