Guide: Understanding size-spectrum dynamics • mizer

This guide gives a concise overview of the core principles of size-spectrum modelling in mizer: physiology, emergent growth and mortality, density dependence, and trophic feedbacks. For the full mathematical formulation, see the model description article.

Mizer models fish communities structured by individual body size rather than age or fixed trophic levels. To diagnose model behaviour you have to reason from size-structured principles, not from age-based or biomass-pool intuition.

What you impose vs. what the model produces

The single most useful habit is knowing which side of the line a quantity sits on: what you set, and what the model works out for itself. This is the distinction that explains most surprises.

What you impose is never an outcome. It is:

The predation process — search volume \(\gamma_i w^q\), the size-preference kernel (beta, sigma), the interaction matrix \(\theta_{ij}\). Together these give a rate of predation events per predator–prey size pair: how fast a predator of size \(w\) would eat prey of size \(w_p\) if it met them.
Individual physiology — maximum intake \(h_i w^n\), metabolic cost \(k_{s,i} w^p\), assimilation efficiency \(\alpha\), and the split between growth and reproduction (w_mat, w_repro_max).
Everything outside predation among the modelled fish — the resource capacity and replenishment rate (kappa, lambda, w_pp_cutoff, r_pp), external mortality z0, fishing effort and selectivity, erepro and R_max.

What the model produces is everything you would actually want to observe: the feeding level \(f_i(w)\), the growth rate \(g_i(w)\) (including whether a fish reaches w_mat at all), the predation mortality \(\mu_p(w)\), the abundance spectra \(N_i(w)\) and their slope, the realised recruitment \(R_i\) and reproduction level \(r_i\), the resource abundance \(N_R(w)\) — and the steady state itself.

Note where the resource sits: you impose the ceiling it grows towards and the speed at which it grows there, but the standing resource abundance is emergent, because the fish eat it. See “The resource is a state variable too” below.

Growth and mortality are one process seen from two ends

The most important consequence, and the one most often missed: growth and predation mortality are not two mechanisms. They are the same predation events, counted twice.

When a predator of size \(w\) eats a prey of size \(w_p\), one event has two bookkeeping consequences: the predator gains mass \(w_p\), and the prey dies. So mizer takes the one pairwise predation rate above and integrates it two different ways:

Emergent quantity	Sum the same predation rate over…	Weighted by	So it responds to
Encounter → feeding level → growth of species \(i\)	all prey small enough for \(i\) to eat	prey mass \(w_p\) (mass gained)	the abundance of \(i\)’s prey — bottom-up
Predation mortality on species \(i\)	all predators big enough to eat \(i\)	nothing (deaths counted, not mass)	the abundance of \(i\)’s predators — top-down

Same kernel \(\phi\), same search volume \(\gamma\), same interaction matrix \(\theta\); even the satiation factor is shared, since a satiated predator both stops gaining mass and stops killing. The only differences are which variable is integrated out — prey size or predator size — and whether you weight by the prey’s mass or just count the death.

This is why you cannot make a species grow faster without making its prey die faster, and why the food and predation feedback loops below are not independent dials. It also means a single diagnosis often covers both symptoms: if growth is wrong, the mortality that same predation imposes is wrong too, and the fix is in the shared parameters, not in one or the other.

Further consequences worth internalising:

You cannot set a growth rate. matchGrowth() adjusts physiological coefficients until the emergent growth matches observation; it does not write a growth curve into the model. Same for mortality. See the guide to reaching steady state and calibrating.
Everything is coupled through abundance. Changing one species’ parameters changes its abundance, which changes the food available to its predators and the mortality on its prey. There are no isolated species.
A steady state is a fixed point of that coupling, not a property you can set species by species. This is why steady() exists.

The size spectrum as state variable

Number density \(N_i(w)\): the state variable for species \(i\) is the density of individuals per unit weight; the number of fish in \([w, w + dw]\) is \(N_i(w)\,dw\).
Biomass density: \(w N_i(w)\). On a logarithmic size grid, the biomass in a bin of constant logarithmic width is \(w N_i(w)\,d(\ln w)\). Plotting choices matter: plotSpectra(params, power = 2) shows biomass per log size bin, which is the view in which the Sheldon spectrum is flat.
Sheldon’s size spectrum: across an aquatic ecosystem, biomass density is roughly uniform across logarithmic size octaves, so abundance scales as \(N(w) \propto w^{-\lambda}\) with \(\lambda \approx 2.05\). In mizer this is an input, not a prediction: lambda sets the resource spectrum the fish feed against. See “The community slope” below.
Community vs species spectra — keep these apart, they behave differently and have different slopes: the combined community spectrum is a steady downward power law, but each species forms a dome spanning from its egg size w_min up to w_max. A species spectrum that looks like a power law over its whole range is a species that is not experiencing the maturity-driven growth slowdown — usually a sign something is wrong.

Bookkeeping in a size bracket

All the dynamics come from one piece of accounting. Pick a size bracket \([w, w + dw]\) and ask what changes the number of fish in it. Individuals do not jump around the size axis: they enter only by growing in through the bottom edge, and leave only by growing out through the top edge or by dying. So

rate of change = what grows in − what grows out − what dies

That is the whole content of the McKendrick–von Foerster equation. The quantity doing the growing-in and growing-out is the flux \(J_i(w)\): the rate

(individuals per year) at which fish of species \(i\) grow past size \(w\). It is just density times speed, \(g_i(w) N_i(w)\) — how many fish are sitting at that size, times how fast they are moving through it. The dying term is \(\mu_i(w) N_i(w)\).

Three things follow, and they cover most of what you need:

The traffic jam effect. If more fish grow in at the bottom of a bracket than grow out at the top, the bracket fills up. That happens wherever growth slows — exactly like traffic on a motorway, where density rises where cars slow down, not where more cars are added. Where growth accelerates, density drops. A bump in a spectrum is usually a growth slowdown, not a recruitment event, and the dome shape of a species spectrum is mostly the growth slowdown at w_repro_max piling fish up before mortality removes them.
Nothing enters except at the egg size. There is no immigration term, so the only way into the spectrum is through the bottom boundary, where the flux is set to the recruitment rate \(R_i(t)\). Everything at every larger size got there by growing.
Steady state is not “nothing happening”. At a fixed point the three terms cancel at every size while fish are still growing and dying at full rate. A steady state can be maintained by fast growth against heavy mortality or by slow growth against light mortality — the balance is what is fixed, not the rates. This is why the ratio of mortality to growth, not either one alone, sets the shape (see below).

With growth diffusion switched on (off by default) individuals of the same size grow at slightly different speeds, which adds a spreading term to the flux; the bookkeeping is unchanged.

The two feedback loops

Close the abundance dependence on the two integrals above — let the prey and predator abundances themselves respond — and each becomes a feedback loop. Nearly every dynamical surprise in mizer comes from one of the two. Both run through body size, which is what makes them different from food-web models.

The food loop (bottom-up, stabilising for the consumer). More consumers → prey depleted → encountered food \(E_i(w)\) falls → feeding level \(f_i(w)\) falls → growth slows → fish spend longer at small sizes where mortality is high → fewer consumers. Diagnose with plotFeedingLevel() and resource_level().

The predation loop (top-down, and the source of cascades). More predators → predation mortality \(\mu_p\) on their prey sizes rises → prey abundance falls → which relieves predation on whatever the prey were eating. Diagnose with plotPredMort() and plotDiet().

The loops act on different sizes of the same species at the same time. A fish is prey early and predator late, so a change can travel up the spectrum through one loop and back down through the other. That round trip is what produces trophic cascades and, when the delay around it matches a generation time, oscillations.

Two knobs decide how tightly the loops are coupled:

beta (preferred predator/prey mass ratio, default 30): a predator eats prey about 30× lighter — roughly 1.5 orders of magnitude down the spectrum. This sets how far apart in size two species have to be before one eats the other.
sigma (kernel width in natural log, default 2): wide. At \(\pm 1\) s.d. the kernel spans mass ratios from about 4 to 200, and it is cut off at ratio 1 (nothing eats prey larger than itself). A narrow kernel couples narrow size bands tightly and is a classic cause of limit cycles; a wide one averages over many prey sizes and damps oscillations.

The resource is a state variable too

The food loop bottoms out in the resource spectrum, and the resource is not a fixed food supply: by default it has dynamics of its own and gets eaten down. What you impose is a capacity \(c_R(w) = \kappa w^{-\lambda}\) (up to w_pp_cutoff, zero above) and a replenishment rate \(r_R(w) = r_{pp} w^{n-1}\); the standing abundance \(N_R(w)\) that fish actually feed on is emergent. The default dynamics (resource_semichemostat()) let each size replenish towards its capacity and be consumed at the rate \(\mu_R(w)\) that the fish impose (getResourceMort()).

Note what is absent: there is no flux along the size axis. Resource sizes do not grow into one another, so a hole eaten in the resource at one size stays at that size and can spread only through fish that feed across it.

The resource level is the mirror of the reproduction level

The dimensionless diagnostic is the resource level \(L(w) = N_R(w)/c_R(w)\) (resource_level()) — how close the resource is running to its ceiling. At the semichemostat steady state \(N_R = r_R c_R/(r_R + \mu_R)\), so \(L = r_R/(r_R + \mu_R)\) and

\[\frac{d \ln N_R}{d \ln \mu_R} = -(1 - L)\]

Read exactly like \(1 - r_i\) for recruitment: \(1 - L\) is the fraction of a change in consumption pressure that actually shows up in food availability.

\(L \to 1\): replenishment overwhelms consumption, the resource sits at its ceiling and barely notices being eaten. The food loop is effectively switched off — an infinite pantry. Species compete for food only weakly, growth does not respond to abundance, and one route to emergent density dependence is gone.
\(L \to 0\): consumption is winning, the standing stock is far below capacity, and every change in consumption passes almost undamped into food availability. Strong competition, strong bottom-up regulation, and a candidate source of oscillations.

Because \(L\) is a ratio at a given abundance, it is the dial to turn after calibration: setResource() with resource_level = 0.5 sets the capacity from the current abundance, leaving the steady state exactly where it was and changing only how stiff the food supply is around it. resource_dynamics = "resource_constant" is the limit \(L \to 1\) taken exactly, and is the clean way to switch the food loop off deliberately when isolating a feedback (see the guide to running a mizer simulation).

Two things that surprise people

A freshly built model has its resource at capacity. newMultispeciesParams() sets \(c_R = \kappa w^{-\lambda}\), which is also the initial abundance, so \(L = 1\) everywhere and the resource is not at a fixed point: replenishment is zero while consumption is not. Project such a model and the resource is drawn down, feeding levels fall, and growth slows — with nothing having been changed. This is normal, and it is why calibration comes before interpretation.
steady() rebalances the resource at the end. It holds the resource fixed while converging the fish, then recomputes the capacity from the (preserved) rate so that the converged state is a steady state of the resource too. The abundance you calibrated against is kept; the capacity moves above it. So after steady() the resource level is an emergent, size-dependent quantity — lowest where the fish feed hardest, near 1 where consumption is negligible.

The resource runs on its own clock

\(r_R \propto w^{n-1}\), the same \(w^{-1/3}\) gradient as everything else, but spread over far more orders of magnitude: with the default r_pp of 10, the smallest resource sizes replenish tens of thousands of times a year, while at the w_pp_cutoff of 10 g the rate is under 5 per year — comparable to fish rates, and precisely at the sizes fish eat most. So the resource is a fast, quasi-static variable at the bottom (which is what getStability() assumes by default; see the guide to analysing dynamic stability) and a genuinely dynamical one at the top. Raising r_pp makes the resource stiffer and the model less sensitive to competition for food; lowering it makes the resource a slow variable that can carry oscillations of its own.

Individual energetics

Energy flows through an individual in three stages.

1 — Encounter and feeding level. A predator of size \(w\) searches volume \(\gamma_i w^q\) and encounters prey according to the predation kernel and the interaction matrix. Encountered food is turned into a dimensionless feeding level \(f_i(w) \in [0,1]\) by a Holling type II response — \(f = 0\) is starvation, \(f = 1\) is complete satiation. Mizer targets \(f_0 \approx 0.6\) by default.

Feeding level is the first thing to look at in almost every diagnosis, because it is dimensionless and has an absolute scale: you can tell at a glance whether a number is wrong. \(f < 0.2\) anywhere in the juvenile range means the species is starving; \(f > 0.9\) everywhere means the species is satiated and effectively decoupled from its prey — its growth will not respond to food very much, and neither will it exert much predation mortality (see below).

2 — Net available energy. Assimilated intake (\(\alpha\), default 0.6) minus standard metabolism, floored at zero. If it hits zero the fish has no surplus for growth or reproduction and simply stops.

3 — Allocation. Below w_mat all net energy goes to somatic growth. Above it, a fraction \(\psi_i(w)\) goes to reproduction and the rest to growth, with \(\psi\) rising to 1 at w_repro_max, so growth halts there. This is what turns the species spectrum into a dome: growth decelerating towards w_repro_max piles individuals up (see the traffic-jam effect below) and then mortality removes them.

What sets the slope

Two different slopes get called “the slope”, and they are not the same object. The species slope is the one that follows from the bookkeeping. The community slope, \(-\lambda\), is a property of the sum over all species and the resource; it is discussed after.

The species slope

This is the bookkeeping above, made quantitative. For one species \(i\), setting “in − out − deaths” to zero and solving gives the local log-log slope of that species’ spectrum, ignoring diffusion:

\[\frac{d \ln N_i}{d \ln w} = -\frac{\mu_i(w)}{g_i(w)/w} - \frac{d \ln g_i}{d \ln w}\]

Read it as: the slope is minus the ratio of mortality to mass-specific growth, minus how fast growth itself is accelerating. With allometric rates \(g \propto w^n\) the second term is just \(n\) (the n species parameter; newMultispeciesParams() sets 2/3 by default).

This one ratio explains most species-spectrum shapes. High mortality or stunted growth ⇒ steep drop, few fish survive to large size. Fast growth relative to mortality ⇒ shallow spectrum. Note that nothing here mentions \(\lambda\): a species’ slope is set by its own mortality and growth, whatever the community around it is doing.

The community slope, and where \(\lambda\) actually comes from

\(\lambda\) is not derived by the model. It is a parameter you set (lambda, default 2.05): the exponent of the resource spectrum the fish feed against. What mizer does is not to predict \(\lambda\) but to keep the model self-consistent with it, in four steps:

Assume the prey a fish sees is distributed as \(N(w) \propto w^{-\lambda}\).
Choose \(q\) to match it. Mizer defaults q to \(\lambda - 2 + n\). Against a \(w^{-\lambda}\) background, a predator searching at rate \(\gamma w^q\) encounters food scaling as \(w^{q + 2 - \lambda}\); that default is exactly the \(q\) making it \(w^n\) — the same power as maximum intake \(h w^n\). Numerator and denominator of the feeding level then scale together, so \(f\) comes out independent of size. (Verify on any fresh model: getFeedingLevel() is flat at f0 across the juvenile range.)
Each species’ slope therefore goes constant. This is the equation above. With \(f\) size-independent, \(g \propto w^n\) and \(\mu \propto w^{n-1}\), so \(\mu/(g/w)\) is constant — a constant slope, i.e. a straight line on log-log axes over each species’ juvenile range.
The community reproduces the assumption. Sum those species spectra — overlapping domes staggered by w_max — together with the resource, and you get back approximately \(w^{-\lambda}\). The loop closes.

Step 4 is exact only in the idealised scaling model, where many species are spread evenly over a wide range of w_max. In a model with a handful of real species it is approximate: the resource carries the small-size end, and the community spectrum is bumpy where species domes overlap unevenly. Do not expect a fitted community slope to come out at exactly lambda.

The practical value of this is diagnostic. The consistency is a chain, and breaking any link shows up in the feeding level first. Change lambda without letting q follow, set q by hand, or deplete the resource away from its assumed power law, and \(f\) acquires a trend in size; species spectra then curve instead of running straight, and the community spectrum stops being a power law. So when a spectrum looks curved, check plotFeedingLevel() for a size trend before touching anything structural.

What sets the timescales

Size sets speed, and this is why age-based intuition misleads.

Mass-specific growth \(g/w \propto w^{n-1} \approx w^{-1/3}\): a fish 1000× heavier grows 10× more slowly in relative terms. Mortality scales the same way (\(\mu \propto w^{n-1}\); mizer’s default external mortality is \(z_0 = 0.6\,w_{\inf}^{\,n-1}\)). So small fish live fast and die young, large fish live slow — and the whole spectrum runs on a gradient of timescales rather than a single one.

Time to grow from egg to w_max scales as \(w_{\max}^{\,1-n} \approx w_{\max}^{1/3}\). A species 1000× larger takes roughly 10× longer to mature. Practical consequences:

Oscillation periods track generation time, so a limit cycle in a large slow species has a long period and needs a correspondingly long project() run to even be visible.
Equilibration is set by the slowest species. A steady() run that looks converged for small species may be nowhere near it for large ones.
Small species and juvenile size classes respond first to any perturbation; the large-fish response arrives a generation later. A transient that looks like a cascade may just be different species running at different speeds.

Density dependence comes in two kinds

This distinction decides how a model responds to fishing, and confusing the two is a common source of wrong conclusions.

1 — Imposed, via R_max. Egg production \(R_{p}\) is converted to realised recruitment \(R\) by a Beverton–Holt relation, summarised by the reproduction level \(r_i = R_i/R_{\max,i} \in [0,1)\) (reproduction_level()) — the fraction of its ceiling that a species’ recruitment is currently running at.

The number that matters is how much of a change in egg production survives into recruitment. For Beverton–Holt that elasticity works out to exactly \[\frac{d \ln R_i}{d \ln R_{p,i}} = 1 - r_i\] so the reproduction level reads directly as the fraction of the response that gets suppressed. This makes the direction easy to keep straight:

Low \(r_i\) (→ 0), weak density dependence: recruitment sits far below its ceiling, so it tracks egg production almost proportionally (\(1 - r_i \approx 1\)). The species responds strongly to anything that changes adult biomass — responsive to fishing, and susceptible to cohort blooms and persistent limit cycles. This is mizer’s default: R_max defaults to \(\infty\), which is \(r_i = 0\) and no imposed density dependence at all.
High \(r_i\) (→ 1), strong density dependence: recruitment is pressed up against \(R_{\max}\), so almost none of a change in egg production gets through (\(1 - r_i \approx 0\)). Fishing that halves adult biomass barely moves recruitment — the stock is buffered. Very stable, and insensitive to fishing in a way that is imposed by assumption rather than earned by the dynamics.

Sanity check if you ever doubt the direction: \(R_{\max} = \infty\) must mean no density dependence, and it gives \(r_i = 0\). Low means weak.

2 — Emergent, via the two feedback loops. With \(R_{\max} = \infty\) a species still is not unregulated: more juveniles deplete the shared resource (food loop) and attract predation (predation loop). This regulation is produced by the model, is size-structured, and spills over onto other species. In a default mizer model it is the only density dependence there is.

How much of it there is at the resource end is set by the resource level: at \(L \to 1\) the shared resource cannot be depleted, so that half of the emergent regulation is missing, whatever R_max says.

Why it matters: both can stabilise a model, but they respond to fishing completely differently. Imposed density dependence is species-local and unmoved by ecosystem change; emergent density dependence weakens when you fish down the predators supplying it. A model made stable by a low R_max is a different ecosystem from one that is stable on its own. If a species seems inert to fishing, check reproduction_level() first — a high value means you are looking at an assumption, not a result.

Size-structured trophic feedbacks

Because trophic role changes with size, mizer shows feedbacks absent from classical food-web models:

Predation release with a starvation bottleneck: fishing large predators lowers \(\mu_p\) on small fish; the surge in small fish overconsumes the resource, feeding levels collapse, and the growth of the predator’s own larvae is stunted. The cascade comes back around and bites the species that started it.
Cultivation–depensation: large predators eat forage fish, but forage fish compete with — or prey on — the larvae of those same predators. Deplete the adults and the forage fish can suppress predator recruitment, holding the predator down even after fishing stops.
Travelling cohort waves: a pulse of recruits eats its way up the size axis, leaving a moving depression in prey abundance and a wake of reduced mortality behind it. Note that the default numerical scheme damps these — see the guide to running a mizer simulation on numerical diffusion before concluding a wave is physically absent.

Diagnostic reasoning heuristics

Check emergent properties before changing structural parameters.

Symptom	Likely cause	What to inspect
Species collapses during `project()`	Starving larvae, intense juvenile predation, or too little egg production	`plotFeedingLevel()`, `plotDiet()`, `getPredMort()`
Biomass oscillates in regular cycles	Reproduction level near 0 (little recruitment damping — mizer’s default), narrow kernel (`sigma` too small), or knife-edge fishing	`reproduction_level()`, `getStability()` (see the guide to analysing dynamic stability)
Growth slows before `w_mat`	Food limitation at intermediate sizes; resource depleted or `w_pp_cutoff` too low	`plotGrowthCurves()`, `resource_level()`, `plotSpectra()`
Feeding levels drift down during `project()` although nothing was changed	The resource was left at its capacity (\(L = 1\), as freshly built) and is being eaten down towards its true fixed point	`resource_level()`, `plot(initialNResource(params))`; run `steady()` first
Growth is not what `matchGrowth()` asked for	Growth is emergent — the food to support it is not there	`plotFeedingLevel()`, then the guide to reaching steady state and calibrating
Tuning one species drops another	Predation overlap or resource competition in shared juvenile size bins	`interaction_matrix()`, `plotDiet()`
Species insensitive to fishing	Strong imposed density dependence (\(r_i\) high, near 1 — only \(1-r_i\) of any change in egg production gets through), or an effectively infinite resource	`reproduction_level()`, `resource_level()`
Species spectrum curves instead of running straight	`lambda`, `q` and `n` no longer mutually consistent, so feeding level is size-dependent	`plotFeedingLevel()` — is it flat in size?
Community slope isn’t `lambda`	Expected only in the idealised scaling model; with few species the domes don’t sum to a clean power law	`getCommunitySlope()`, `plotSpectra()` — compare against species domes, not against `lambda`
Species won’t stay at steady state	Fixed point is dynamically unstable, not a numerical failure	`getSteadyResidual()`, `steadyNewton()`

Reference formulas

The equations behind the reasoning above, for when you need the exact form.

Dynamics — the McKendrick–von Foerster equation: the “in − out − deaths” balance above, written as a continuity equation in size.

\[\frac{\partial N_i(w,t)}{\partial t} + \frac{\partial J_i(w,t)}{\partial w} = -\mu_i(w,t) N_i(w,t)\]

The derivative \(\partial J_i/\partial w\) is the net loss by growth: flux out at the top of the bracket minus flux in at the bottom. The right-hand side is the loss by death. Growth flux \(J_i = g_i N_i\), boundary condition \(J_i(w_{\min}, t) = R_i(t)\) at the egg size, \(g_i\) in \(\text{g}\cdot\text{year}^{-1}\) and \(\mu_i\) in \(\text{year}^{-1}\). With growth diffusion switched on (D_ext > 0, off by default) the flux gains a term \(-\tfrac{1}{2}\,\partial(D_i N_i)/\partial w\), which smooths sharp peaks and spreads cohorts.

The two integrals of the same predation rate. Encountered food — integrate over prey size \(w_p\), weighting by the mass \(w_p\) gained:

\[E_i(w) = \gamma_i(w) \int \Big(\theta_{ip} N_R(w_p) + \sum_j \theta_{ij} N_j(w_p)\Big)\, \phi_i(w/w_p)\, w_p\, dw_p\]

Predation mortality — integrate the same thing over predator size \(w'\), counting deaths rather than mass:

\[\mu_{p,i}(w) = \sum_j \int \theta_{ji}\,\big(1 - f_j(w')\big)\,\gamma_j (w')^q\,\phi_j(w'/w)\,N_j(w')\,dw'\]

The shared factors \(\gamma\), \(\phi\) and \(\theta\) are what makes these two sides of one coin. The \((1 - f_j)\) factor is worth noting: satiated predators kill less, and a species whose predators are all near \(f = 1\) experiences less mortality than their abundance alone suggests. It applies to the growth side too — realised intake is \(f_i h_i w^n = (1 - f_i) E_i\) — so satiation throttles both integrals together.

Feeding level from encountered food \(E_i\) and maximum intake \(h_i w^n\):

\[f_i(w) = \frac{E_i(w)}{E_i(w) + h_i w^n}\]

Net energy after assimilation (\(\alpha\)) and metabolism (\(k_s w^p\), plus an activity term \(k w\) that defaults to zero):

\[E_{\text{net},i}(w) = \max\big(0,\; \alpha_i f_i(w) h_i w^n - k_{s,i} w^p\big)\]

Growth, after the reproductive allocation \(\psi_i(w)\):

\[g_i(w) = \big(1 - \psi_i(w)\big) E_{\text{net},i}(w)\]

Predation kernel (lognormal, the default), for mass ratio \(\text{ppmr} = w/w_p \ge 1\) and zero below:

\[\phi_i(\text{ppmr}) = \exp\!\left[\frac{-\big(\ln(\text{ppmr}/\beta_i)\big)^2}{2\sigma_i^2}\right]\]

Resource dynamics — the default semichemostat, applied to each size independently (no transport along the size axis):

\[\frac{\partial N_R(w,t)}{\partial t} = r_R(w)\big[c_R(w) - N_R(w,t)\big] - \mu_R(w,t) N_R(w,t)\]

with replenishment rate \(r_R(w) = r_{pp} w^{n-1}\), capacity \(c_R(w) = \kappa w^{-\lambda}\) below w_pp_cutoff and zero above, and \(\mu_R\) the consumption by the fish. The steady state is \(N_R = r_R c_R/(r_R + \mu_R)\), i.e. resource level \(L = N_R/c_R = r_R/(r_R + \mu_R)\). resource_logistic() replaces the bracket by \(N_R(1 - N_R/c_R)\) and resource_constant() holds \(N_R\) fixed.

Egg production and Beverton–Holt density dependence:

\[R_{p,i} = \frac{\epsilon_i}{2 w_{\min}} \int \psi_i(w)\, E_{\text{net},i}(w)\, N_i(w)\, dw, \qquad R_i = \frac{R_{\max,i} R_{p,i}}{R_{\max,i} + R_{p,i}}\]

Defaults worth remembering: beta 30, sigma 2, lambda 2.05, w_pp_cutoff 10 g, kappa 1e11, r_pp 10, resource_dynamics "resource_semichemostat", alpha 0.6, f0 0.6, n 2/3, p 0.7, q \(= \lambda - 2 + n\), erepro 1, R_max \(\infty\), w_min 0.001 g, w_mat \(= w_{\inf}/4\), z0 \(= 0.6\,w_{\inf}^{\,n-1}\).

For the complete mathematical formulation see the model description article.