sim: coexistence

[willvieira]

Calculate invasion growth rate ($\lambda$) for pair of competing species in two different conditions:

• Invasion growth rate without competition: $\lambda_{i,N_j = 0}$
• Invasion growth rate with competing species at equilibrium: $\lambda_{i,N_j^\ast}$

Both lambdas are then used to calculate the sensitivity ($S_i$) of the invading species $i$ to the resident species $j$:

$$ Si = \frac{\lambda{i,Nj = 0} - \lambda{i,Nj^\ast}}{\lambda{i,N_j = 0}} $$

With the sensitivity of both species ($S_i$ and $S_j$) to the other, we can finally compute the two coexistence metrics: niche difference (ND) and relative fitness difference (RFD) following Carroll et al. 2011 and Narwani et al. 2013

$$ ND = 1 - \sqrt{S_i S_j} $$

$$ RFD = \sqrt{\frac{S_i}{S_j}} $$

Note that both ND and RFD can be computed with total population size ($N^\ast$) or total biomass ($BA^\ast$) instead of the invasion growth rate. Also note that both ND and RFD are in function of temperature and precipitation set to the optima value of the resident species $j$. Finally, they are also dependent on plot random effects that are defined to zero for the moment.

Output

• $\lambda_{i,Nj = 0}$, $N{i, Nj = 0}^\ast$, and $BA{i, N_j = 0}^\ast$
• $\lambda_{i,Nj^\ast}$, $N{i,Nj^\ast}^\ast$, and $BA{i,N_j^\ast}^\ast$
• $\lambda_{j,Ni=0}$, $N{j,Ni=0}^\ast$, and $BA{j,N_i=0}^\ast$

Questions:

• What is the distribution of sensitivity ($S_i$) across the interaction matrix?
• How the species pair are distributed in the coexistence graph of ND x RFD?
• How does RFD relates to $\frac{\lambda_{i,Nj = 0}}{\lambda{j,N_i = 0}}$
• Similarly, following Chesson 2000, 2018 derivation from the LV model, how does RFD relates to $\frac{K{i}}{K{j}}$ where $K$ is the carrying capacity $\frac{1}{ki} = \alpha{ii}$, and $\alpha{ij} = \frac{N{ij}^\ast}{N_{i}^\ast}$

[willvieira]

Sensitivity of invader species i over resident speciesj at equilibrium within its optimal environment. I computed the sensitivity using three different metrics: i) initial lambda, ii) population size at equilibrium (N), and iii) population biomass at equilibrium (BA).

[willvieira]

Using BA among the three metrics (reason to be described), next figure shows the distribution of BA sensitivity across species and their shade tolerance.

[willvieira]

BA sensitivity can be seen as the interspecific competition parameter $\alpha{ij}$. Using the following formula, we can compare the BA sensitivity to $\alpha{ij}$:

$$ \alpha{ij} = log( \frac{BA{i,Nj^\ast}}{BA{i,N_j=0}}) $$

[willvieira]

Using the carrying capacity ($K = BA_{i,Nj = 0}$) to define $\alpha{ii} = 1/K$: