class: center, middle, inverse, title-slide # TD2 - Movement Ecology models ### Marie-Pierre Etienne ### October, 2023 --- # Four classical movement models We denote by `\(X_k\)` the position at time `\(t_k\)` et `\(V_k\)` the velocity. ## Random walk on `\(\mathbb{R}^2\)` `$$X_{k+1} - X_k = \varepsilon_{k+1}, \quad (\varepsilon_{k}) \overset{i.i.d}{\sim}\mathcal{N}(0,\sigma^2)$$` ## Auto regressive Model on `\(\mathbb{R}^2\)` - Central place behaviour `$$X_{k+1} - X_k = \rho (X_k - \mu) + \varepsilon_{k+1}, \quad (\varepsilon_{k}) \overset{i.i.d}{\sim}\mathcal{N}(0,\sigma^2)$$` ## Integrated Random walk `$$V_{k+1}- V_k = \varepsilon_{k+1},\quad X_{k+1} = X_k + V_{k+1} + \nu_{k+1}, \quad (\varepsilon_{k}) \overset{i.i.d}{\sim}\mathcal{N}(0,\sigma^2), \quad (\nu_{k}) \overset{i.i.d}{\sim}\mathcal{N}(0,\tau^2)$$` ## Integrated Correlated Random walk `$$V_{k+1} - V_k = - \rho(V_k-\mu) + \varepsilon_{k+1}, X_{k+1} = X_k + V_{k+1} + \nu_{k+1}, \quad (\varepsilon_{k}) \overset{i.i.d}{\sim}\mathcal{N}(0,\sigma^2)$$` --- # Two classical movement models - simulation Write 2 functions to simulate this movement model, and compare for the same level of noise, starting from `\((0,0)\)` the first time the distance to `\((0,0)\)` exceeds `\(25\)`, Change the value of `\(\mu\)` to measure its impact. --- # How to integrate the landscape ? * A natural barrier * A jaguar in a cage, how to deal with the limits of the cage ? What does it change for the stationnary distribution ? -- .pull-left[ * a spatiallly structure covariate] .pull-right[ ``` ## grf: simulation on a set of locations provided by the user ## grf: process with 1 covariance structure(s) ## grf: nugget effect is: tausq= 0 ## grf: covariance model 1 is: exponential(sigmasq=10, phi=8) ## grf: simulation using the function GaussRF from package RandomFields ## grf: End of simulation procedure. Number of realizations: 1 ``` <!-- --> ]