Ontogenetic symmetry and asymmetry in energetics

Abstract

Body size ($\equiv$ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

Appendix: The explicit expressions for \(\gamma _i\) and \(\alpha _i\) and \(\frac{d\tilde{R}}{dm}\)

In this appendix we give the missing ingredients of formulas (48a) and (48b).

$$\begin{aligned} \gamma _{0}&= \dfrac{G(\tilde{R})\left(\nu _{A}(\tilde{R}) -d_{A}-m\right)\left(I_{J}(\tilde{R})-I_{A} (\tilde{R})\right)}{\left(I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R}) -d_{A}-m\right)-I_{A}(\tilde{R}) \left(\nu _{J}(\tilde{R})-d_{J}-m\right)\right)^{2}}\nonumber \\&-\dfrac{G(\tilde{R})}{I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R}) -d_{A}-m\right)-I_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)}\nonumber \\ \gamma _{1}&= \dfrac{G^{\prime }(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)+G(\tilde{R})\nu ^{\prime }_{A}(\tilde{R})}{I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)-I_{A} (\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right) }\nonumber \\&-\dfrac{G(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)^{2}I^{{\prime }}_{J}(\tilde{R})}{\left(I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)-I_{A}(\tilde{R}) \left(\nu _{J}(\tilde{R})-d_{J}-m\right)\right)^{2}}\nonumber \\&+\dfrac{G(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)I^{\prime }_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)}{\left(I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)-I_{A}(\tilde{R}) \left(\nu _{J}(\tilde{R})-d_{J}-m\right)\right)^{2}}\nonumber \\&-\dfrac{G(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)\left(I_{J}(\tilde{R})\nu ^{\prime }_{A}(\tilde{R})-I_{A}(\tilde{R})\nu ^{\prime }_{J}(\tilde{R})\right)}{\left(I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)-I_{A} (\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)\right)^{2}} \end{aligned}$$

(60)

$$\begin{aligned} \alpha _{0}&= \dfrac{G(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)\left(I_{A}(\tilde{R})-I_{J}(\tilde{R})\right)}{\left(I_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)-I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)\right)^{2}}\nonumber \\&-\dfrac{G(\tilde{R})}{I_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)-I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)}\nonumber \\ \alpha _{1}&= \dfrac{G^{\prime }(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)+G(\tilde{R})\nu ^{\prime }_{J}(\tilde{R})}{I_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)-I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)} \nonumber \\&-\dfrac{G(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)^{2}I^{{\prime }}_{A}(\tilde{R})}{\left(I_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)-I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)\right)^{2}}\nonumber \\&+\dfrac{G(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)I^{\prime }_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)}{\left(I_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)-I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)\right)^{2}}\nonumber \\&-\dfrac{G(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)\left(I_{A}(\tilde{R})\nu ^{\prime }_{J}(\tilde{R})-I_{J}(\tilde{R})\nu ^{\prime }_{A}(\tilde{R})\right)}{\left(I_{A}(\tilde{R})\left(\nu _{J}(\tilde{R})-d_{J}-m\right)-I_{J}(\tilde{R})\left(\nu _{A}(\tilde{R})-d_{A}-m\right)\right)^{2}} \end{aligned}$$

(61)

Determining the derivative $d\tilde{R}(m)/dm$ using the implicit function theorem after substitution of $d_{J}+m$ and $d_{A}+m$ for $d_{J}$ and $d_{A},$ respectively, in Eq. (44) leads to

$$\begin{aligned} \dfrac{\nu _{A}(\tilde{R})z^{(d_{J}+m)/\nu _{J}(\tilde{R})-1}}{d_{A}+m}\left( \dfrac{\nu ^{\prime }_{A}(\tilde{R})}{\nu _{A}(\tilde{R})}-\dfrac{(d_{J}+m)\ln (z)\nu ^{\prime }_{J}(\tilde{R})}{\left(\nu _{J}(\tilde{R})\right)^{2}} \right)\dfrac{d\tilde{R}}{dm}(m)\nonumber \\ +\dfrac{\nu _{A}(\tilde{R})z^{(d_{J}+m)/\nu _{J}(\tilde{R})-1}}{d_{A}+m}\left( \dfrac{\ln (z)}{\nu _{J}(\tilde{R})}-\dfrac{1}{d_{A}+m} \right)=0\quad \Rightarrow \nonumber \\ \dfrac{d\tilde{R}}{dm}(m)= \left(\dfrac{1}{d_{A}\!+\!m}-\dfrac{\ln (z)}{\nu _{J}(\tilde{R})}\right)\!\!\! \left(\dfrac{\nu ^{\prime }_{A}(\tilde{R})}{\nu _{A}(\tilde{R})}-\dfrac{(d_{J}+m)\ln (z)\nu ^{\prime }_{J}(\tilde{R})}{\left(\nu _{J}(\tilde{R})\right)^{2}}\right)^{\!\!\!-1} \end{aligned}$$

(62)