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 ... ... @@ -21,7 +21,7 @@ using ABMEv ## Parameters of the simulation Parameters are stored in the parameter dictionary p ### General parameters - "reflected"=>true: if true then reflection occurs on the first trait -which should stand for geographic position. Depending on the agent type, reflections occurs in the domain $ [-1,1] $ or between nodes 1 and p["nodes"] - "reflected"=>true: if true then reflection occurs on the first trait -which should stand for geographic position. Depending on the agent type, reflections occurs in the domain  [-1,1]  or between nodes 1 and p["nodes"] - "alpha" => α: is the competition function - "K" => K: is the birth rate - "tend" => 1.5: is the time to end simulation ... ... @@ -30,16 +30,16 @@ Parameters are stored in the parameter dictionary p ### Mutation If anisotropy in mutation, the following parameters should be declared as arrays where each entry corresponds to a dimension. - mu The probability of mutation. - D If mutation happens on the agent, the increment follows $\mathcal{N}_{ 0, D}$ - D If mutation happens on the agent, the increment follows \mathcal{N}_{ 0, D} ### Birth #### Growth - K is the birth coefficient ( $b(x) = K(x)$ ) - K is the birth coefficient ( b(x) = K(x) ) ### Death #### Competition - Competition between agent with trait x and y is defined as α(x,y) - Death coefficient is defined as $d(x^{(i)}) = \sum_j^{N(t)} \alpha(x^{(i)},x^{(j)})$ - Death coefficient is defined as d(x^{(i)}) = \sum_j^{N(t)} \alpha(x^{(i)},x^{(j)}) ### Fitness Fitness is defined internally as b - d. ... ...
 ... ... @@ -7,24 +7,24 @@ ### Rates Each individual is assigned a birth $ b_i $ and death $ d_i $ rate. The total rate is given by the sum of all individual rates Each individual is assigned a birth  b_i  and death  d_i  rate. The total rate is given by the sum of all individual rates math R(t) = \left[ \sum_i b_i(t) + d_i(t) \right]  A particular event, birth or death, is chosen at random with a probability equal to the rate of this event divided by the total rate $R$ A particular event, birth or death, is chosen at random with a probability equal to the rate of this event divided by the total rate R > This has to be checked, we are still not hundred percent sure ### 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) $. > 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). 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) . > 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). #### Inversion method 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 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 math P(I_F(U) \leq x ) = P(U \leq F(x)) = F(x)  Thanks to the ***inversion method*** we get the incremental time step $dt$, exponentially distributed with parameter $\lambda = R(t)$, as Thanks to the ***inversion method*** we get the incremental time step dt, exponentially distributed with parameter \lambda = R(t), as math dt(\omega) = -\frac{\log(U(\omega))}{R(t)} \iff X(\omega) = \exp(-U(t)dt(\omega)) ... ...
 ... ... @@ -5,7 +5,7 @@ The Wright Fisher process is an individual based model where the number of agent ![alt text](https://upload.wikimedia.org/wikipedia/commons/0/0b/Random_sampling_genetic_drift.svg) At each time step, $N$ agents are picked up from previous generation to reproduce. Their number of offspring is proportional to their fitness, calculated as usual with **birth and death rates**. At each time step, N agents are picked up from previous generation to reproduce. Their number of offspring is proportional to their fitness, calculated as usual with **birth and death rates**. It takes thus **only one time step to go trough one generation**. Thus it is more suit- able for numerical simulations. In practice, the Moran and Wright–Fisher models give qualitatively similar results, but genetic drift runs twice as fast in the Moran model. ... ...
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 ... ... @@ -33,7 +33,7 @@ Which has Hessian matrix \frac{\partial^2 f}{\partial y_i \partial y_j}(x,y) = -w_i(x)w_j(x) + w_j(x) x_i^3 - 2 x_i^3x_j^3 - \left(3 x_i^2 - \frac{1}{\sigma_i^2} \right) \delta_{ij}  This Hessian matrix can possess positive eigenvalues depending on the variance of the Gaussian components $\sigma_i$ and the coordinates in the phenotypic space. This Hessian matrix can possess positive eigenvalues depending on the variance of the Gaussian components \sigma_i and the coordinates in the phenotypic space. ## References - [Evolutionary dynamics from deterministic microscopic ecological processes](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.101.032411) ... ...
 ... ... @@ -2,25 +2,25 @@ # Equivalence with PDE models > HDR Champagnat Theorem 2.5 Section 2.2 > Recall that the IBM is a Markov process in $\mathcal{M}^1$ which state at time $t$ is the measure > Recall that the IBM is a Markov process in \mathcal{M}^1 which state at time t is the measure math \nu_t = \sum_{i=1}^{N_t} \delta_{x_i}  > Hence an individual with trait $x$ in population $\nu_t$ gives birth to a new individual at rate $b(x)$ and dies at rate > Hence an individual with trait x in population \nu_t gives birth to a new individual at rate b(x) and dies at rate math d(x) + \int c(x,y)(\nu_t(dy) - \delta_x(dy)) = d(x) - c(x,x) + \sum_{i=1}^{N_t} c(c,x_i)  The Dirac mass in the integral stands for the fact that an individual is not in competition with itself. Hence when $N_t = 1$ the competition term cancels. When there is birth, with probability $p(x)$ the offspring has trait $y = x + H$ where $H$ is a random variablee with law $m(x,h)\,dh$. The Dirac mass in the integral stands for the fact that an individual is not in competition with itself. Hence when N_t = 1 the competition term cancels. When there is birth, with probability p(x) the offspring has trait y = x + H where H is a random variablee with law m(x,h)\,dh. ## Assumptions Assume that #### Assumption 1 : bounds and parameters regularity. - Functions $b,c,d,p$ are continuous and bounded, with positive are nul values - $\exists \bar{m}, \forall x \in D, h \in \R^d m(x,h) \leq \bar{m}(h)$ - Functions b,c,d,p are continuous and bounded, with positive are nul values - \exists \bar{m}, \forall x \in D, h \in \R^d m(x,h) \leq \bar{m}(h) #### Assumption 2 $\nu_t^K = \frac{1}{K}\nu_t$ where $\nu_t$ is constructed such that $c\equiv \frac{1}{K}c$ \nu_t^K = \frac{1}{K}\nu_t where \nu_t is constructed such that c\equiv \frac{1}{K}c ... ... @@ -33,7 +33,7 @@ Then we get the **large population limit without mutation scaling** ## Mutations Assuming a small mutational variance $\max \sigma^2_i << 1$ and a mutation rate $U$, the mutational effects can be approximated by an elliptic operator $\sum_{i=1}^{n} (\mu_i^2/x)\partial_{ii}$ with $\mu_i = \sigma_i\sqrt{U}$ Assuming a small mutational variance \max \sigma^2_i << 1 and a mutation rate U, the mutational effects can be approximated by an elliptic operator \sum_{i=1}^{n} (\mu_i^2/x)\partial_{ii} with \mu_i = \sigma_i\sqrt{U} > :warning: check that with Burger In other words (from Champagnat, Ferriere and Meleard 2006), we have ... ...
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