Dynamics of feedbacks in nonequilibrium biodiversity organizational scale¶

Carlos J. Melian, Eawag, ETH-Domain, Switzerland¶

---

https://github.com/melian009/Ecoevon/tree/master/CCSS2024¶

References¶

Rainey, P., Travisano, M. (1998). Adaptive radiation in a heterogeneous environment. Nature 394:69–72.¶

Kleidon, A. (2010). Non-equilibrium thermodynamics, maximum entropy production and Earth-system evolution. Philo. Trans. Math., Phys., and Eng. Sciences, 368:181-196.¶

Melo and Marroig (2015). Directional selection can drive the evolution of modularity in complex traits. PNAS, 112:470-475.¶

Goswami et al (2015). The fossil record of phenotypic integration and modularity: A deep-time perspective on developmental and evolutionary dynamics. PNAS, 112:4891-4896.¶

Laughlin D.C., and Messier J. (2015). Fitness of multidimensional phenotypes in dynamic adaptive landscapes. TREE, 30:487-496.¶

Barros, C. et al. (2016). N‐dimensional hypervolumes to study stability of complex ecosystems. Ecol. Lett., 19:729–742.¶

Boyle et al. (2017). An expanded view of complex traits: From polygenic to omnigenic. Cell. 169:1177-1186.¶

Melián et al. (2018). Deciphering the interdependence between ecological and evolutionary networks. TREE, 33:504-512.¶

Andreazzi, C., et al (2024). Biodiversity dynamics with complex genotype-to-phenotype architectures. Submitted to Biol. Rev.¶

Where are we now¶

Nonequilibrium¶

Feedbacks¶

Where are we gonna go¶

Biodiversity organizational scale¶

Route to dimensionality¶

Where are we now¶

Nonequilibrium¶

Feedback¶

Where are we gonna go¶

${\bf \Omega_{BAM}}$ = $\begin{bmatrix} V_B & C_{BA} & C_{BM} \\ C_{AB} & V_A & C_{AM} \\ C_{MB} & C_{MA} & V_M \end{bmatrix}$

$\mathbf{W}({\bf z})^{t}_{ix} = exp[-\gamma({\color{red}[{\bf z}^{t}_{ix} - {\bf \theta^{t}}_{ix}\color{red}]^T} {\bf \Omega_{BAM}}^{-1} {\color{red}[{\bf z}^{t}_{ix} - {\bf \theta^{t}}_{ix}\color{red}]})]$

$ \underbrace{\begin{bmatrix} W({\bf z_{B}}^{t}_{ix}) \\ W({\bf z_{A}}^{t}_{ix}) \\ \vdots \\ W({\bf z_{M}}^{t}_{ix}) \\ \end{bmatrix} }_{\mathbf{W}}$ = $\underbrace{\begin{bmatrix} W({B}^{t}_{ix})^{*} \\ W({A}^{t}_{ix})^{*} \\ \vdots \\ W({M}^{t}_{ix})^{*} \\ \end{bmatrix}^{T} }_{\mathbf{W}}$ $\underbrace{\begin{bmatrix} V_{B} & C_{BA} & \dots & C_{BM} \\ C_{AB} & V_{A} & \dots & C_{AM} \\ \vdots & \vdots & \vdots & \vdots \\ C_{MB} & C_{MA} & \dots & V_{M} \\ \end{bmatrix}^{-1} }_{\mathbf{\Omega_{BAM}}}$ $\underbrace{\begin{bmatrix} W({B}^{t}_{ix})^{*} \\ W({A}^{t}_{ix})^{*} \\ \vdots \\ W({M}^{t}_{ix})^{*} \\ \end{bmatrix} }_{\mathbf{W}}$

\begin{equation} {\Large \partial_t\rho (\bf{z},t) = -\nabla_{\bf{z}} \big[\nabla_{\bf{z}} F(\bf{z},\bf{y_t})\rho(\bf{z,t})\big]} {\Large -r(\bf{z},\bf{y_t})\rho(\bf{z,t})} \nonumber \\ {\Large+\int_{\Omega}\int_{\Omega}M(\bf{z}|\bf{z'},\bf{z''})B(\bf{z'},\bf{z''})\rho(\bf{z'},t)\rho(\bf{z''},t)d^{d}\bf{z'}d^{d}\bf{z''}} \end{equation}

Biodiversity organizational scale¶

Route to Dimensionality¶

Complex traits: GPA¶

\begin{equation} {\Large {z}^{i}_{j}}(x,t) = {\Large {f}(\bf{L})}(x,t), {\Large {f}:\mathbb{R}^{m} \rightarrow \mathbb{R}^{z}} \end{equation}

In this formulation, an individual’s $i$ phenotype in site $x$ and time $t$ is given by

\begin{equation} {\Large \bf{Z}^{i}}(x,t) = {\Large {f}(\bf{L}) = D[{f} (\bf{L})] = \bf{B} \bf{Y}}, \end{equation}\begin{equation} {\Large \bf{Z}^{i}}(x,t) = {\Large \bf{B}^{T} \bf{E} \bf{Y}} \end{equation}

Abiotic trait¶

\begin{equation} {\Large D({z}^{i}_{a})}(x,t) = {\Large \lvert 0.5 - cdf(\mathcal{N}(\theta_{a}, \sigma^2), z^{i}_{a}) \lvert}, \end{equation}

where $D({z}^{i}_{a})$ is the distance of abiotic trait of individual $i$ to its optimum, $\theta_{a}$ is the optimal value used as the mean of a normal distribution, $\sigma^{2}$ is the variance of the normal distribution, and $z^{i}_{a}$ is the value of the abiotic trait $i$ and $cdf$ is cumulative distribution function. The fitness of the abiotic trait of individual $i$ is then given by

\begin{equation} {\Large W(z^{i}_{a})}(x,t) = {\Large 1 - D({z}^{i}_{a})}(x,t) \end{equation}

Biotic trait¶

\begin{equation} {\Large D({z}^{i}_{b} z^{j}_{b})}(x,t) = {\Large \lvert 0.5 - cdf(\mathcal{N}(z^{i}_{b}, \sigma^2), z^{j}_{b}) \lvert}. \end{equation}

The strength of an interaction is a function of species-species coefficient and the phenotypic distance between the two individuals

\begin{equation} {\Large s_{z^{i}_{b} z^{j}_{b}}}(x,t) = {\Large (1 - D({z}^{i}_{b} z^{j}_{b})}(x,t)) \times {\Large \lvert c_{z^{A}_{b} z^{B}_{b}} \lvert} \times {\Large sign(c_{z^{A}_{b} z^{B}_{b}})} \end{equation}

Fitness¶

\begin{equation} {\Large W({Z}^{i})}(x,t) = {\Large W({z}^{i}_{a})}(x,t) + {\Large s_{z^{i}_{b} z^{j}_{b}}}(x,t), \end{equation}

and the fitness function takes into account the selection coefficient as

\begin{equation} {\Large W(Z^{i})}(x,t) = {\Large 1 - ( (1 - W(Z^{i})(x,t)) \times s_A)} \end{equation}

Take home message¶

Where are we now¶

• gap in understanding bb-ba-ab-aa-interactions accounting for nonequilibrium and feedbacks at many spatiotemporal scales

Where are we gonna go¶

• GPA connecting complex traits to biotic-abiotic feedbacks and diversity patterns
• The route to dimensionality integrating BOS to feedbacks