We propose that any individual have a constant birth rate, and competes with all the individuals present in the same patch. Assume there are ``N_t`` individuals at time ``t``.
Let ``i \in \{ 1,2,\dots,N_t\}``. ``x_{i,t} \in \{1,2,\dots,9\}`` denotes the position of the ``i``-th individual at time ``t``.
Let ``i \in \{ 1,2,\dots,N_t\}``. ``x_{i} \in \{1,2,\dots,9\}`` denotes the position of the ``i``-th individual.
The competition pressure experienced by individual ``i`` is such that
@@ -16,7 +16,7 @@ A particular event, birth or death, is chosen at random with a probability equal
### Time steps
An event is exponentiallly distributed in time, with parameter ``\lambda = U(t)``. This makes events memoryless, meaning that the probability of having a birth or death event is always the same, no matter when (``P(X > s_t | X > t) = P(X > s) ``.
!!! tip "Inversion method"
<!-- !!! tip "Inversion method"
Let ``B(t) = \sum_i b_i(t)`` and ``D(t) = \sum_i d_i(t)``. Let ``T_b, T_d`` the time for a birth or death event to occur. Then we have ``P(T_b < T_d) = \frac{B(t)}{B(t) + D(t)}`` (competing exponentials).
Let ``U`` be an ``\mathcal{U}_{(0,1)}``-distributed random variable and ``F \colon \R \to [0,1]`` be a distribution function. Then we have
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@@ -29,7 +29,7 @@ An event is exponentiallly distributed in time, with parameter ``\lambda = U(t)`