TABLE 2.

Model equations for each species' ecological dynamics

Equation	Description
Species population dynamics	$\frac{\partial N\left(m\right)}{\partial t}+\frac{\partial }{\partial m}(g(m)N(m))=‐\mu (m)N(m)$	E1
Background resource dynamics	$\frac{\partial N_R(m,t)}{\partial t}=r_0{m}^{p‐1}[{\kappa }_R{m}^{‐\lambda }‐N_R(m,t)]‐{\mu }_p(m)N_R(m,t)$	E2
Prey size selection by size m predator	$\varphi (m,m_p)=\exp \left[\frac{‐{(\ln (m/(\beta m_p)))}^2}{2{\sigma }^2}\right]$	E3
Encountered food by size m predator across all sizes (mp) of species j prey	$E(m)=\gamma m^q\int (N_R(m_p)+{\sum }_j{\theta }_{j}N(m_p))\varphi (m,m_p)m_p\mathrm{d}{m}_p$	E4
Feeding level	$f(m)=\frac{E(m)}{E(m)+h{m}^n}$	E5
Energy allocation toward reproduction	$\psi \left(m\right)={\left[1+{\left(\frac{m}{m^{\ast }}\right)}^{‐u}\right]}^{‐1}{\left(\frac{m}{M}\right)}^{1‐n}$	E6
Somatic growth	$g(m)=(\alpha f(m)h{m}^n‐k_s{m}^p)(1‐\psi (m))$	E7
Reproduction	$R_p=\frac{\epsilon }{2m_0}\int N(m)(\alpha f(m)h{m}^n‐k_s{m}^p)\psi (m)\mathrm{d}m$	E8
Recruitment	$R=R_{max}\frac{R_p}{R_p+R_{max}}$	E9
Predation mortality on size mp prey inflicted by all sizes (m) of species j predators	${\mu }_p(m_p)={\sum }_j\int \varphi (m,m_p)(1‐f_i(m)){\gamma }_j{m}^q{\theta }_j{N}_j(m)dm$	E10
Senescence mortality	${\mu }_{\mathit{se}}=\left\{\begin{array}{cc}s{e}_{min}+(s{e}_{max}‐s{e}_{min})e^{‐\frac{M}{m}}& \text{if}m>m\ast \\ 0,& \text{otherwise}\end{array}\right)$	E11
Starvation mortality	${\mu }_{\mathit{st}}=\left\{\begin{array}{cc}s{t}_{r}m(k_s{m}^p‐\alpha f(m)h{m}^n),& \text{if}{k}_s{m}^p>\alpha f(m)h{m}^n\\ 0,& \text{otherwise}\end{array}\right)$	E12
Background mortality	${\mu }_b={\mu }_0{M}^{n‐1}$	E13
Fishing mortality	${\mu }_f=\left\{\begin{array}{cc}0.8,& \text{if}m\ge m\ast \\ 0,& \text{otherwise}\end{array}\right)$	E14

Subscripts for each species are not included in the equations below for readability. These dynamics also hold for each phenotype (nested within each species) once they have entered the size spectrum. Equation numbers reference the processes illustrated in Figure 1 and descriptions in main text. N(m) is the density at size driven by: g(m) which is the feeding dependent growth rate at size and μ(m) which is the mortality at size, the latter is comprised of several mortality terms below. All parameter values and definitions are provided in Table 3 or the main text.