From 7181201dffbf55091d3d4baee176745aeb1f565f Mon Sep 17 00:00:00 2001
From: Gustav Delius <gustav.delius@gmail.com>
Date: Thu, 11 Jun 2026 18:21:45 +0100
Subject: [PATCH] Second-order plotting: draw bin averages at geometric bin
 centres (#382)

Once a quantity is a finite-volume cell average N_j it no longer lives at the
left bin edge w_j but at the geometric bin centre w*_j = sqrt(w_j w_{j+1}) =
w_j sqrt(beta) (the log-midpoint, exact for the community spectrum w^-2).
Plotting it at the bin boundary mis-places it by half a bin.

This adds a `representation` tag ("point"/"average") to the
ArraySpeciesBySize and ArrayTimeBySpeciesBySize classes, threaded through the
constructors, subset/slice methods and the MizerSim rate path
(sim_size_rate / get_species_size_rate_from_sim). The central size-axis helpers
get_ArraySpeciesBySize_w() and the new get_ArrayTimeBySpeciesBySize_w() return
the geometric bin centres (new bin_midpoints() helper) for "average" arrays,
but only when the model uses second-order bin-averaging
(second_order_w[["bin_average"]]) -- so default plots, as.data.frame() output
and vdiffr snapshots are unchanged.

The bin-averaged sinks getPredMort/getFMort/getMort/getExtMort and the
reproductive investment getERepro tag themselves "average"; point-valued
quantities (encounter, growth) stay "point" and on the nodes. Everything
downstream (prepare_*_plot_data, as.data.frame, plot2) follows the central
helpers automatically.

plotSpectra/plotCDF build their own data frames and handle the w^power weight;
their centre placement is the separate #383. Snapshots gained only the new
benign `representation` attribute (accepted). Full suite: 2824 pass.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
---
 NEWS.md                                       | 10 ++++
 R/ArraySpeciesBySize-class.R                  | 27 +++++++--
 R/ArrayTimeBySpeciesBySize-class.R            | 53 ++++++++++++++---
 R/helpers.R                                   | 27 +++++++++
 R/plots.R                                     |  6 +-
 R/rate_functions.R                            | 43 +++++++++-----
 R/setExtMort.R                                |  6 +-
 man/ArraySpeciesBySize.Rd                     | 14 ++++-
 man/ArrayTimeBySpeciesBySize.Rd               | 14 ++++-
 man/bin_midpoints.Rd                          | 31 ++++++++++
 man/get_ArraySpeciesBySize_w.Rd               |  8 ++-
 man/get_ArrayTimeBySpeciesBySize_w.Rd         | 23 ++++++++
 man/get_species_size_rate_from_sim.Rd         |  3 +-
 man/sim_size_rate.Rd                          |  1 +
 .../_snaps/backwards_compatibility.md         |  6 ++
 tests/testthat/_snaps/project_methods.md      | 45 +++++++++++++++
 tests/testthat/test-ArraySpeciesBySize.R      | 57 +++++++++++++++++++
 .../testthat/test-ArrayTimeBySpeciesBySize.R  | 31 ++++++++++
 tests/testthat/test-project_methods.R         |  3 +-
 vignettes/numerical_details.qmd               |  2 +
 20 files changed, 375 insertions(+), 35 deletions(-)
 create mode 100644 man/bin_midpoints.Rd
 create mode 100644 man/get_ArrayTimeBySpeciesBySize_w.Rd

diff --git a/NEWS.md b/NEWS.md
index da2aafb09..8dabcf37f 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -1,5 +1,15 @@
 # mizer (development version)
 
+- Size-resolved diagnostics that are finite-volume bin averages (the
+  mortalities `getPredMort()`, `getFMort()`, `getMort()`, `getExtMort()`, and
+  the reproductive investment `getERepro()`) are now drawn at the geometric bin
+  centre `sqrt(w_j w_{j+1})` rather than the left bin edge, the location where a
+  bin average actually lives. The `ArraySpeciesBySize`/`ArrayTimeBySpeciesBySize`
+  classes carry a `representation` tag (`"point"`/`"average"`) recording this,
+  and the shift is applied only when the model uses second-order bin-averaging
+  (`second_order_w[["bin_average"]]`), so default plots are unchanged.
+  Point-valued quantities (encounter, growth) stay on the grid nodes. (#382)
+
 - `plotYield()` now uses `sim2 = NULL` instead of `missing(sim2)` to detect
   the optional second simulation argument. This is backward-compatible and
   makes the function work correctly with `do.call()`.
diff --git a/R/ArraySpeciesBySize-class.R b/R/ArraySpeciesBySize-class.R
index c0c39976b..b28247f6b 100644
--- a/R/ArraySpeciesBySize-class.R
+++ b/R/ArraySpeciesBySize-class.R
@@ -23,6 +23,11 @@
 #' @param units A string giving the units (e.g. "g/year", "1/year").
 #' @param params A `MizerParams` object. Used for species colours, linetypes,
 #'   and size ranges in the `plot()` method.
+#' @param representation Either `"point"` (the default) for a quantity sampled
+#'   at the grid nodes, or `"average"` for a finite-volume bin average. A
+#'   bin-averaged quantity is drawn at the geometric bin centre rather than the
+#'   left bin edge, but only when the model uses second-order bin-averaging
+#'   (`second_order_w[["bin_average"]]`), so default plots are unchanged.
 #'
 #' @return An `ArraySpeciesBySize` object (inherits from `matrix` and `array`).
 #' @seealso [print()], [summary()], [as.data.frame()], [plot()]
@@ -34,10 +39,12 @@
 #' summary(enc)
 #' }
 ArraySpeciesBySize <- function(x, value_name = NULL, units = NULL,
-                               params = NULL) {
+                               params = NULL,
+                               representation = c("point", "average")) {
     if (!is.matrix(x)) {
         stop("`x` must be a matrix.")
     }
+    representation <- match.arg(representation)
     if (!is.null(params) && identical(dim(x), dim(params@metab))) {
         dimnames(x) <- dimnames(params@metab)
     }
@@ -45,7 +52,8 @@ ArraySpeciesBySize <- function(x, value_name = NULL, units = NULL,
         class = c("ArraySpeciesBySize", "matrix", "array"),
         value_name = value_name,
         units = units,
-        params = params
+        params = params,
+        representation = representation
     )
 }
 
@@ -860,7 +868,13 @@ as.data.frame.ArraySpeciesBySize <- function(x, row.names = NULL,
 #'
 #' @param x An `ArraySpeciesBySize` object.
 #'
-#' @return A numeric vector giving the size represented by each column.
+#' @return A numeric vector giving the size represented by each column. When the
+#'   array is tagged as a bin average (`representation = "average"`) *and* the
+#'   model uses second-order bin-averaging (`second_order_w[["bin_average"]]`),
+#'   the geometric bin centres are returned instead of the left bin edges, so
+#'   that bin-averaged quantities are drawn at the size where they actually live
+#'   (see [bin_midpoints()]). Point-valued quantities and first-order models are
+#'   unaffected, keeping default plots unchanged.
 #' @keywords internal
 get_ArraySpeciesBySize_w <- function(x) {
     params <- attr(x, "params")
@@ -871,11 +885,13 @@ get_ArraySpeciesBySize_w <- function(x) {
         }
         return(w)
     }
+    average <- identical(attr(x, "representation"), "average") &&
+        isTRUE(params@second_order_w[["bin_average"]])
     if (ncol(x) == length(params@w)) {
-        return(params@w)
+        return(if (average) bin_midpoints(params) else params@w)
     }
     if (ncol(x) == length(params@w_full)) {
-        return(params@w_full)
+        return(if (average) bin_midpoints(params, w_full = TRUE) else params@w_full)
     }
     stop("Can not determine the size grid for this ArraySpeciesBySize object. ",
          "The number of columns is ", ncol(x), ", but the params object has ",
@@ -891,6 +907,7 @@ get_ArraySpeciesBySize_w <- function(x) {
         attr(result, "value_name") <- attr(x, "value_name")
         attr(result, "units") <- attr(x, "units")
         attr(result, "params") <- attr(x, "params")
+        attr(result, "representation") <- attr(x, "representation")
         class(result) <- c("ArraySpeciesBySize", "matrix", "array")
     }
     result
diff --git a/R/ArrayTimeBySpeciesBySize-class.R b/R/ArrayTimeBySpeciesBySize-class.R
index 8d0a8053a..7112556e9 100644
--- a/R/ArrayTimeBySpeciesBySize-class.R
+++ b/R/ArrayTimeBySpeciesBySize-class.R
@@ -25,6 +25,11 @@
 #' @param units A string giving the units (e.g. "1/year").
 #' @param params A `MizerParams` object. Used for species colours, linetypes,
 #'   and size ranges in the `plot()` and `animateSpectra()` methods.
+#' @param representation Either `"point"` (the default) for a quantity sampled
+#'   at the grid nodes, or `"average"` for a finite-volume bin average, which is
+#'   then drawn at the geometric bin centre when the model uses second-order
+#'   bin-averaging (`second_order_w[["bin_average"]]`). See
+#'   [ArraySpeciesBySize()].
 #'
 #' @return An `ArrayTimeBySpeciesBySize` object (inherits from `array`).
 #' @seealso [print()], [summary()], [as.data.frame()], [plot()],
@@ -38,18 +43,46 @@
 #' plot(fmort, time = 2007)
 #' }
 ArrayTimeBySpeciesBySize <- function(x, value_name = NULL, units = NULL,
-                                     params = NULL) {
+                                     params = NULL,
+                                     representation = c("point", "average")) {
     if (!is.array(x) || length(dim(x)) != 3) {
         stop("`x` must be a 3D array.")
     }
+    representation <- match.arg(representation)
     structure(x,
         class = c("ArrayTimeBySpeciesBySize", "array"),
         value_name = value_name,
         units = units,
-        params = params
+        params = params,
+        representation = representation
     )
 }
 
+#' Get the size grid for an ArrayTimeBySpeciesBySize object
+#'
+#' Internal helper, the three-dimensional analogue of
+#' [get_ArraySpeciesBySize_w()]. Returns the geometric bin centres (see
+#' [bin_midpoints()]) when the array is tagged as a bin average and the model
+#' uses second-order bin-averaging, otherwise the grid nodes read from the size
+#' dimension names. Falls back to the dimension names when no `params` is
+#' attached.
+#'
+#' @param x An `ArrayTimeBySpeciesBySize` object.
+#' @return A numeric vector giving the size represented by each size slice.
+#' @keywords internal
+get_ArrayTimeBySpeciesBySize_w <- function(x) {
+    w <- as.numeric(dimnames(x)[[3]])
+    if (any(is.na(w))) w <- seq_len(dim(x)[3])
+    params <- attr(x, "params")
+    average <- !is.null(params) &&
+        identical(attr(x, "representation"), "average") &&
+        isTRUE(params@second_order_w[["bin_average"]])
+    if (!average) return(w)
+    if (dim(x)[3] == length(params@w)) return(bin_midpoints(params))
+    if (dim(x)[3] == length(params@w_full)) return(bin_midpoints(params, w_full = TRUE))
+    w
+}
+
 #' Test if an object is an ArrayTimeBySpeciesBySize
 #'
 #' @param x An object to test.
@@ -230,6 +263,7 @@ ArrayTimeBySpeciesBySize_slice <- function(x, time = NULL) {
     params <- attr(x, "params")
     value_name <- attr(x, "value_name")
     units <- attr(x, "units")
+    representation <- attr(x, "representation") %||% "point"
 
     times <- as.numeric(dimnames(x)[[1]])
     if (is.null(time)) {
@@ -243,7 +277,8 @@ ArrayTimeBySpeciesBySize_slice <- function(x, time = NULL) {
                     nrow = dim(arr)[2],
                     dimnames = dimnames(arr)[2:3])
     ArraySpeciesBySize(slice, value_name = value_name,
-                       units = units, params = params)
+                       units = units, params = params,
+                       representation = representation)
 }
 
 #' @rdname plotHover
@@ -316,7 +351,7 @@ animate.ArrayTimeBySpeciesBySize <- function(x, species = NULL,
         times <- times[times <= tlim[2]]
     }
 
-    w <- as.numeric(dimnames(arr)[[3]])
+    w <- get_ArrayTimeBySpeciesBySize_w(x)
     sub <- arr[, species, , drop = FALSE]
 
     # Build long-format data frame; time varies fastest to match c(sub)
@@ -366,8 +401,7 @@ as.data.frame.ArrayTimeBySpeciesBySize <- function(x, row.names = NULL,
     times <- as.numeric(dimnames(x)[[1]])
     if (any(is.na(times))) times <- seq_len(dim(x)[1])
     sp_names <- dimnames(x)[[2]]
-    w <- as.numeric(dimnames(x)[[3]])
-    if (any(is.na(w))) w <- seq_len(dim(x)[3])
+    w <- get_ArrayTimeBySpeciesBySize_w(x)
 
     data.frame(
         expand.grid(time = times, Species = sp_names, w = w,
@@ -385,17 +419,20 @@ as.data.frame.ArrayTimeBySpeciesBySize <- function(x, row.names = NULL,
         attr(result, "value_name") <- attr(x, "value_name")
         attr(result, "units") <- attr(x, "units")
         attr(result, "params") <- attr(x, "params")
+        attr(result, "representation") <- attr(x, "representation")
         class(result) <- c("ArrayTimeBySpeciesBySize", "array")
     } else if (is.matrix(result)) {
         dim_names <- names(dimnames(result))
         attrs <- list(value_name = attr(x, "value_name"),
                       units = attr(x, "units"),
-                      params = attr(x, "params"))
+                      params = attr(x, "params"),
+                      representation = attr(x, "representation") %||% "point")
         if (identical(dim_names, c("sp", "w"))) {
             result <- ArraySpeciesBySize(result,
                                          value_name = attrs$value_name,
                                          units = attrs$units,
-                                         params = attrs$params)
+                                         params = attrs$params,
+                                         representation = attrs$representation)
         } else if (identical(dim_names, c("time", "sp"))) {
             result <- ArrayTimeBySpecies(result,
                                          value_name = attrs$value_name,
diff --git a/R/helpers.R b/R/helpers.R
index 30b657c87..88461f7b4 100644
--- a/R/helpers.R
+++ b/R/helpers.R
@@ -66,6 +66,33 @@ power_law_bin_average <- function(w, dw, d, w_max = Inf) {
     }
 }
 
+#' Geometric bin centres of the size grid
+#'
+#' Internal helper for the second-order plotting code. A finite-volume cell
+#' average \eqn{N_j = (1/\Delta w_j)\int_{w_j}^{w_{j+1}} N\,dw} does not live at
+#' the left bin edge \eqn{w_j} but at the geometric bin centre
+#' \deqn{w^*_j = \sqrt{w_j\,w_{j+1}} = w_j\sqrt{\beta},}
+#' where \eqn{\beta = w_{j+1}/w_j} is the (constant) bin ratio of the
+#' logarithmic grid. This is the log-symmetric, second-order-correct location at
+#' which to plot a bin-averaged quantity (it is exact for the community spectrum
+#' \eqn{N\propto w^{-2}}). It is a uniform half-bin shift to the right on the log
+#' axis, the same for the consumer grid `w` and the full prey/resource grid
+#' `w_full`.
+#'
+#' @param params A MizerParams object.
+#' @param w_full If `TRUE`, return the centres of the full (resource) grid
+#'   `params@w_full`; otherwise the consumer grid `params@w`.
+#' @return A numeric vector of geometric bin centres, one per grid node.
+#' @concept helper
+#' @keywords internal
+bin_midpoints <- function(params, w_full = FALSE) {
+    # The grid is geometric, so the bin ratio beta is constant across the whole
+    # (consumer and resource) grid.
+    beta <- params@w_full[2] / params@w_full[1]
+    w <- if (w_full) params@w_full else params@w
+    w * sqrt(beta)
+}
+
 #' Length-weight conversion
 #'
 #' For each species, convert between length and weight using the relationship
diff --git a/R/plots.R b/R/plots.R
index b69d81731..ca6ff584c 100644
--- a/R/plots.R
+++ b/R/plots.R
@@ -2539,7 +2539,8 @@ plotPredMort.MizerSim <- function(object, species = NULL,
     pred_mort <- ArraySpeciesBySize(pred_mort,
                                     value_name = "Predation mortality",
                                     units = "1/year",
-                                    params = object@params)
+                                    params = object@params,
+                                    representation = "average")
     plot(pred_mort, species = species, all.sizes = all.sizes,
          highlight = highlight, wlim = wlim, llim = llim,
          log_x = log_x, log_y = log_y, log = log,
@@ -2674,7 +2675,8 @@ plotFMort.MizerSim <- function(object, species = NULL,
         f <- apply(f, c(2, 3), mean)
     }
     f <- ArraySpeciesBySize(f, value_name = "Fishing mortality",
-                            units = "1/year", params = object@params)
+                            units = "1/year", params = object@params,
+                            representation = "average")
     plot(f, species = species, all.sizes = all.sizes,
          highlight = highlight, wlim = wlim, llim = llim,
          log_x = log_x, log_y = log_y, log = log,
diff --git a/R/rate_functions.R b/R/rate_functions.R
index b78098824..f3f5ca9b4 100644
--- a/R/rate_functions.R
+++ b/R/rate_functions.R
@@ -200,7 +200,8 @@ get_sim_rate_slice <- function(sim, time_idx) {
 #' @keywords internal
 get_species_size_rate_from_sim <- function(sim, time_range, drop,
                                            rate_fun, value_name,
-                                           units = NULL) {
+                                           units = NULL,
+                                           representation = "point") {
     time_elements <- get_sim_rate_time_elements(sim, time_range)
 
     # Apply the one-time rate calculation to each selected saved time. The
@@ -220,11 +221,13 @@ get_species_size_rate_from_sim <- function(sim, time_range, drop,
         result <- ArrayTimeBySpeciesBySize(result,
                                           value_name = value_name,
                                           units = units,
-                                          params = sim@params)
+                                          params = sim@params,
+                                          representation = representation)
     } else if (is.matrix(result) &&
                names(dimnames(result))[[1]] == "sp") {
         result <- ArraySpeciesBySize(result, value_name = value_name,
-                                     units = units, params = sim@params)
+                                     units = units, params = sim@params,
+                                     representation = representation)
     }
     result
 }
@@ -409,7 +412,8 @@ mizer_rates_subset <- function(params, n, n_pp, n_other, t, effort,
 #' @keywords internal
 sim_size_rate <- function(sim, time_range, drop, target, slot,
                           value_name, units = NULL,
-                          use_sim_effort = FALSE, ...) {
+                          use_sim_effort = FALSE,
+                          representation = "point", ...) {
     params <- validParams(sim@params)
     rates_fns <- projectRateFunctions(params)
     effort <- params@initial_effort
@@ -435,7 +439,8 @@ sim_size_rate <- function(sim, time_range, drop, target, slot,
             }
             m
         },
-        value_name = value_name, units = units)
+        value_name = value_name, units = units,
+        representation = representation)
 }
 
 #' Build a `MizerSim` rate getter that resolves the rate functions once
@@ -758,8 +763,11 @@ getPredMort.MizerParams <- function(object, n, n_pp, n_other,
         pred_mort <- f(params, n = n, n_pp = n_pp, n_other = n_other, t = t,
                        pred_rate = pred_rate)
     }
+    # Predation mortality is a sink integrated against the prey density over the
+    # prey bin, so under second-order bin-averaging it is a bin average and is
+    # plotted at the geometric bin centre (see get_ArraySpeciesBySize_w()).
     ArraySpeciesBySize(pred_mort, value_name = "Predation mortality",
-             units = "1/year", params = params)
+             units = "1/year", params = params, representation = "average")
 }
 
 #' @export
@@ -768,7 +776,7 @@ getPredMort.MizerSim <- function(object, n, n_pp, n_other,
     sim <- object
     sim_size_rate(sim, time_range, drop, target = "PredMort",
                   slot = "pred_mort", value_name = "Predation mortality",
-                  units = "1/year", ...)
+                  units = "1/year", representation = "average", ...)
 }
 
 #' Alias for `getPredMort()`
@@ -1119,7 +1127,8 @@ getFMort.MizerParams <- function(object, effort, time_range, drop = TRUE,
                                     n_other = n_other, time_range = t))
         fmort <- do.call(f, c(args, list(...)))
         fmort <- ArraySpeciesBySize(fmort, value_name = "Fishing mortality",
-                           units = "1/year", params = params)
+                           units = "1/year", params = params,
+                           representation = "average")
         return(fmort)
     } else if (length(effort) == no_gears) {
         args <- list(
@@ -1131,7 +1140,8 @@ getFMort.MizerParams <- function(object, effort, time_range, drop = TRUE,
                                     n_other = n_other, time_range = t))
         fmort <- do.call(f, c(args, list(...)))
         fmort <- ArraySpeciesBySize(fmort, value_name = "Fishing mortality",
-                           units = "1/year", params = params)
+                           units = "1/year", params = params,
+                           representation = "average")
         return(fmort)
     } else {
         stop("Invalid effort argument")
@@ -1144,7 +1154,7 @@ getFMort.MizerSim <- function(object, effort, time_range, drop = TRUE,
     sim <- object
     sim_size_rate(sim, time_range, drop, target = "FMort", slot = "f_mort",
                   value_name = "Fishing mortality", units = "1/year",
-                  use_sim_effort = TRUE, ...)
+                  use_sim_effort = TRUE, representation = "average", ...)
 }
 
 
@@ -1218,7 +1228,8 @@ getMort.MizerParams <- function(object,
                              t = t, effort = effort,
                              rates_fns = rates_fns, targets = "Mort", ...)
     return(ArraySpeciesBySize(r$mort, value_name = "Total mortality",
-                     units = "1/year", params = params))
+                     units = "1/year", params = params,
+                     representation = "average"))
 }
 
 #' @export
@@ -1228,7 +1239,7 @@ getMort.MizerSim <- function(object, n, n_pp, n_other, effort, t, ...,
     if (missing(time_range) && !missing(t)) time_range <- t
     sim_size_rate(sim, time_range, drop, target = "Mort", slot = "mort",
                   value_name = "Total mortality", units = "1/year",
-                  use_sim_effort = TRUE, ...)
+                  use_sim_effort = TRUE, representation = "average", ...)
 }
 
 #' Alias for `getMort()`
@@ -1292,8 +1303,11 @@ getERepro.MizerParams <- function(object, n = initialN(object),
         erepro <- f(params, n = n, n_pp = n_pp, n_other = n_other, t = t,
                     e = e)
     }
+    # The per-capita reproductive investment is the integrand of the
+    # reproduction integral, where (under second-order bin-averaging) it enters
+    # as a bin average, so it is plotted at the geometric bin centre.
     ArraySpeciesBySize(erepro, value_name = "Energy for reproduction",
-             units = "g/year", params = params)
+             units = "g/year", params = params, representation = "average")
 }
 
 #' @export
@@ -1302,7 +1316,8 @@ getERepro.MizerSim <- function(object, n, n_pp, n_other, t, ...,
     sim <- object
     if (missing(time_range) && !missing(t)) time_range <- t
     sim_size_rate(sim, time_range, drop, target = "ERepro", slot = "e_repro",
-                  value_name = "Energy for reproduction", units = "g/year", ...)
+                  value_name = "Energy for reproduction", units = "g/year",
+                  representation = "average", ...)
 }
 
 #' Alias for `getERepro()`
diff --git a/R/setExtMort.R b/R/setExtMort.R
index 8a4acddfe..57e6c5afe 100644
--- a/R/setExtMort.R
+++ b/R/setExtMort.R
@@ -178,10 +178,14 @@ getExtMort <- function(params) {
 }
 #' @export
 getExtMort.MizerParams <- function(params) {
+    # External mortality is a sink integrated against the abundance over the
+    # bin; under second-order bin-averaging mu_b is the exact bin average
+    # (see setExtMort()), so it is plotted at the geometric bin centre.
     ArraySpeciesBySize(params@mu_b,
                        value_name = "External mortality",
                        units = "1/year",
-                       params = params)
+                       params = params,
+                       representation = "average")
 }
 
 #' @rdname setExtMort
diff --git a/man/ArraySpeciesBySize.Rd b/man/ArraySpeciesBySize.Rd
index 8bb7d6369..f30bdd2ac 100644
--- a/man/ArraySpeciesBySize.Rd
+++ b/man/ArraySpeciesBySize.Rd
@@ -4,7 +4,13 @@
 \alias{ArraySpeciesBySize}
 \title{S3 class for species x size rate arrays}
 \usage{
-ArraySpeciesBySize(x, value_name = NULL, units = NULL, params = NULL)
+ArraySpeciesBySize(
+  x,
+  value_name = NULL,
+  units = NULL,
+  params = NULL,
+  representation = c("point", "average")
+)
 }
 \arguments{
 \item{x}{A matrix (species x size).}
@@ -15,6 +21,12 @@ ArraySpeciesBySize(x, value_name = NULL, units = NULL, params = NULL)
 
 \item{params}{A \code{MizerParams} object. Used for species colours, linetypes,
 and size ranges in the \code{plot()} method.}
+
+\item{representation}{Either \code{"point"} (the default) for a quantity sampled
+at the grid nodes, or \code{"average"} for a finite-volume bin average. A
+bin-averaged quantity is drawn at the geometric bin centre rather than the
+left bin edge, but only when the model uses second-order bin-averaging
+(\code{second_order_w[["bin_average"]]}), so default plots are unchanged.}
 }
 \value{
 An \code{ArraySpeciesBySize} object (inherits from \code{matrix} and \code{array}).
diff --git a/man/ArrayTimeBySpeciesBySize.Rd b/man/ArrayTimeBySpeciesBySize.Rd
index b06c11c40..ff4205f29 100644
--- a/man/ArrayTimeBySpeciesBySize.Rd
+++ b/man/ArrayTimeBySpeciesBySize.Rd
@@ -4,7 +4,13 @@
 \alias{ArrayTimeBySpeciesBySize}
 \title{S3 class for time x species x size arrays}
 \usage{
-ArrayTimeBySpeciesBySize(x, value_name = NULL, units = NULL, params = NULL)
+ArrayTimeBySpeciesBySize(
+  x,
+  value_name = NULL,
+  units = NULL,
+  params = NULL,
+  representation = c("point", "average")
+)
 }
 \arguments{
 \item{x}{A 3D array (time x species x size).}
@@ -15,6 +21,12 @@ ArrayTimeBySpeciesBySize(x, value_name = NULL, units = NULL, params = NULL)
 
 \item{params}{A \code{MizerParams} object. Used for species colours, linetypes,
 and size ranges in the \code{plot()} and \code{animateSpectra()} methods.}
+
+\item{representation}{Either \code{"point"} (the default) for a quantity sampled
+at the grid nodes, or \code{"average"} for a finite-volume bin average, which is
+then drawn at the geometric bin centre when the model uses second-order
+bin-averaging (\code{second_order_w[["bin_average"]]}). See
+\code{\link[=ArraySpeciesBySize]{ArraySpeciesBySize()}}.}
 }
 \value{
 An \code{ArrayTimeBySpeciesBySize} object (inherits from \code{array}).
diff --git a/man/bin_midpoints.Rd b/man/bin_midpoints.Rd
new file mode 100644
index 000000000..1bd5038d8
--- /dev/null
+++ b/man/bin_midpoints.Rd
@@ -0,0 +1,31 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/helpers.R
+\name{bin_midpoints}
+\alias{bin_midpoints}
+\title{Geometric bin centres of the size grid}
+\usage{
+bin_midpoints(params, w_full = FALSE)
+}
+\arguments{
+\item{params}{A MizerParams object.}
+
+\item{w_full}{If \code{TRUE}, return the centres of the full (resource) grid
+\code{params@w_full}; otherwise the consumer grid \code{params@w}.}
+}
+\value{
+A numeric vector of geometric bin centres, one per grid node.
+}
+\description{
+Internal helper for the second-order plotting code. A finite-volume cell
+average \eqn{N_j = (1/\Delta w_j)\int_{w_j}^{w_{j+1}} N\,dw} does not live at
+the left bin edge \eqn{w_j} but at the geometric bin centre
+\deqn{w^*_j = \sqrt{w_j\,w_{j+1}} = w_j\sqrt{\beta},}
+where \eqn{\beta = w_{j+1}/w_j} is the (constant) bin ratio of the
+logarithmic grid. This is the log-symmetric, second-order-correct location at
+which to plot a bin-averaged quantity (it is exact for the community spectrum
+\eqn{N\propto w^{-2}}). It is a uniform half-bin shift to the right on the log
+axis, the same for the consumer grid \code{w} and the full prey/resource grid
+\code{w_full}.
+}
+\concept{helper}
+\keyword{internal}
diff --git a/man/get_ArraySpeciesBySize_w.Rd b/man/get_ArraySpeciesBySize_w.Rd
index 8a072f195..01f8e5aa4 100644
--- a/man/get_ArraySpeciesBySize_w.Rd
+++ b/man/get_ArraySpeciesBySize_w.Rd
@@ -10,7 +10,13 @@ get_ArraySpeciesBySize_w(x)
 \item{x}{An \code{ArraySpeciesBySize} object.}
 }
 \value{
-A numeric vector giving the size represented by each column.
+A numeric vector giving the size represented by each column. When the
+array is tagged as a bin average (\code{representation = "average"}) \emph{and} the
+model uses second-order bin-averaging (\code{second_order_w[["bin_average"]]}),
+the geometric bin centres are returned instead of the left bin edges, so
+that bin-averaged quantities are drawn at the size where they actually live
+(see \code{\link[=bin_midpoints]{bin_midpoints()}}). Point-valued quantities and first-order models are
+unaffected, keeping default plots unchanged.
 }
 \description{
 Internal helper that returns the consumer size grid \code{params@w} or the full
diff --git a/man/get_ArrayTimeBySpeciesBySize_w.Rd b/man/get_ArrayTimeBySpeciesBySize_w.Rd
new file mode 100644
index 000000000..185ab6d94
--- /dev/null
+++ b/man/get_ArrayTimeBySpeciesBySize_w.Rd
@@ -0,0 +1,23 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/ArrayTimeBySpeciesBySize-class.R
+\name{get_ArrayTimeBySpeciesBySize_w}
+\alias{get_ArrayTimeBySpeciesBySize_w}
+\title{Get the size grid for an ArrayTimeBySpeciesBySize object}
+\usage{
+get_ArrayTimeBySpeciesBySize_w(x)
+}
+\arguments{
+\item{x}{An \code{ArrayTimeBySpeciesBySize} object.}
+}
+\value{
+A numeric vector giving the size represented by each size slice.
+}
+\description{
+Internal helper, the three-dimensional analogue of
+\code{\link[=get_ArraySpeciesBySize_w]{get_ArraySpeciesBySize_w()}}. Returns the geometric bin centres (see
+\code{\link[=bin_midpoints]{bin_midpoints()}}) when the array is tagged as a bin average and the model
+uses second-order bin-averaging, otherwise the grid nodes read from the size
+dimension names. Falls back to the dimension names when no \code{params} is
+attached.
+}
+\keyword{internal}
diff --git a/man/get_species_size_rate_from_sim.Rd b/man/get_species_size_rate_from_sim.Rd
index f1a492db2..78d706ad7 100644
--- a/man/get_species_size_rate_from_sim.Rd
+++ b/man/get_species_size_rate_from_sim.Rd
@@ -10,7 +10,8 @@ get_species_size_rate_from_sim(
   drop,
   rate_fun,
   value_name,
-  units = NULL
+  units = NULL,
+  representation = "point"
 )
 }
 \arguments{
diff --git a/man/sim_size_rate.Rd b/man/sim_size_rate.Rd
index 3a06319f4..1e7ee2dcc 100644
--- a/man/sim_size_rate.Rd
+++ b/man/sim_size_rate.Rd
@@ -13,6 +13,7 @@ sim_size_rate(
   value_name,
   units = NULL,
   use_sim_effort = FALSE,
+  representation = "point",
   ...
 )
 }
diff --git a/tests/testthat/_snaps/backwards_compatibility.md b/tests/testthat/_snaps/backwards_compatibility.md
index c935db0a2..11aa3bc45 100644
--- a/tests/testthat/_snaps/backwards_compatibility.md
+++ b/tests/testthat/_snaps/backwards_compatibility.md
@@ -1991,6 +1991,8 @@
         Haddock 156333250686 140753274052 126067041926 112316282311  99528337355
         Cod       1859436226   1297418578    906990223    640148493    460469325
         Saithe     106982064     75453886     57866920     48355351     43425648
+      attr(,"representation")
+      [1] "point"
 
 # set_trait_model() works as in version 1
 
@@ -4235,6 +4237,8 @@
         9  0.01326478 0.01316974 0.01307207 0.01297159 0.01286811 0.01276144
         10 0.01326478 0.01316974 0.01307207 0.01297159 0.01286811 0.01276144
         11 0.01326478 0.01316974 0.01307207 0.01297159 0.01286811 0.01276144
+      attr(,"representation")
+      [1] "point"
 
 # set_community_model() works as in version 1
 
@@ -4752,4 +4756,6 @@
                  w
       sp            811000    1e+06
         Community 3539.996 3503.139
+      attr(,"representation")
+      [1] "point"
 
diff --git a/tests/testthat/_snaps/project_methods.md b/tests/testthat/_snaps/project_methods.md
index b4f2d8bd8..b53f386b4 100644
--- a/tests/testthat/_snaps/project_methods.md
+++ b/tests/testthat/_snaps/project_methods.md
@@ -8,6 +8,11 @@
           "attributes": {},
           "value": [3, 100]
         },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["point"]
+        },
         "dimnames": {
           "type": "list",
           "attributes": {
@@ -90,6 +95,11 @@
           "type": "character",
           "attributes": {},
           "value": ["1/year"]
+        },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["point"]
         }
       },
       "value": [1.14752828e-10, 7.69233583e-07, 5.16467603e-14, 7.46511953e-11, 1.06515634e-06, 5.53705909e-14, 1.46390086e-10, 1.47213888e-06, 2.8970455e-14, 1.37707922e-10, 2.03078552e-06, 5.39136062e-14, 1.60514742e-10, 2.79614241e-06, 2.93587784e-14, 1.53408612e-10, 3.8426749e-06, 5.70878716e-14, 1.95406033e-10, 5.27091744e-06, 4.95424293e-14, 1.56063416e-10, 7.21631356e-06, 5.76256677e-14, 1.3895232e-10, 9.86096475e-06, 2.75090857e-14, 1.31602487e-10, 0.00001345, 3.72871968e-14, 1.65180536e-10, 0.00001831, 2.55534635e-14, 1.69778775e-10, 0.00002487, 3.3636702e-14, 6.40351108e-11, 0.00003373, 1.04299851e-15, 8.17832111e-11, 0.00004565, 4.17199404e-14, 1.01841897e-10, 0.00006167, 0, 9.53648511e-11, 0.00008314, 2.39889658e-14, 6.53006502e-11, 0.00011186, 6.25799107e-15, 1.29129904e-10, 0.00015022, 1.87739732e-14, 1.04566535e-10, 0.00020132, 6.25799107e-15, 1.00751742e-10, 0.00026926, 0, 8.84694394e-11, 0.00035942, 0, 1.05453207e-10, 0.00047878, 0, 4.73014329e-11, 0.00063648, 0, 9.25684775e-11, 0.00084439, 0, 5.49533183e-11, 0.00111789, 0, 8.94385835e-11, 0.00147688, 0, 1.66739059e-11, 0.00194706, 2.50319643e-14, 3.99152586e-11, 0.00256149, 0, 0, 0.00336262, 0, 0, 0.00440481, 0, 0, 0.00575752, 0, 0, 0.00750922, 0, 0, 0.00977233, 3.33759524e-14, 0, 0.01268935, 0, 0, 0.01644035, 0, 0, 0.02125231, 0, 0, 0.02741059, 0, 0, 0.03527286, 0, 0, 0.04528627, 1.33503809e-13, 0, 0.05800822, 3.33759524e-14, 0, 0.07413156, 1.66879762e-13, 0, 0.09451507, 0, 2.1281133e-10, 0.1202201, 0, 4.58314406e-10, 0.15255454, 6.67519047e-14, 1.21104643e-09, 0.19312534, 0, 2.36645932e-09, 0.24390104, 6.67519047e-14, 4.75600646e-09, 0.30728586, 2.67007619e-13, 8.65942422e-09, 0.38620732, 0, 1.50202466e-08, 0.48421916, 2.00255714e-13, 2.50722824e-08, 0.60562208, 6.67519047e-14, 3.99336595e-08, 0.75560452, 3.33759524e-13, 6.13770413e-08, 0.9404062, 2.67007619e-13, 9.08808492e-08, 1.16750737, 0, 1.30175826e-07, 1.44584671, 0, 1.81009805e-07, 1.78607108, 2.00255714e-13, 2.45783717e-07, 2.20082051, 4.00511428e-13, 3.27090741e-07, 2.70505173, 4.67263333e-13, 4.3093748e-07, 3.31640365, 9.34526666e-13, 5.645011e-07, 4.05560814, 2.03593309e-12, 7.414353e-07, 4.94694904, 4.07186619e-12, 9.79922233e-07, 6.01877242, 8.11035642e-12, 1.30569716e-06, 7.30405027, 1.5386314e-11, 1.75204846e-06, 8.84099946, 2.91372064e-11, 2.3635429e-06, 10.6737569, 5.38020352e-11, 3.19718768e-06, 12.85311094, 9.82588037e-11, 4.33350375e-06, 15.43728769, 1.76058149e-10, 5.88787258e-06, 18.4927898, 3.11264132e-10, 8.04129544e-06, 22.09528329, 5.41341259e-10, 0.00001107, 26.33052634, 9.27183957e-10, 0.00001543, 31.29533163, 1.56369674e-09, 0.00002176, 37.09855172, 2.59906051e-09, 0.00003105, 43.86207398, 4.2583877e-09, 0.00004465, 51.7218093, 6.88116182e-09, 0.00006444, 60.82865542, 1.09713514e-08, 0.00009288, 71.34941312, 1.72695313e-08, 0.00013318, 83.46763024, 2.68508869e-08, 0.00018953, 97.38434554, 4.12632573e-08, 0.00026739, 113.31870165, 6.27138609e-08, 0.00037405, 131.50839376, 9.43279197e-08, 0.00051944, 152.2099182, 1.40499342e-07, 0.00071731, 175.69858362, 2.07369419e-07, 0.00098697, 202.2682459, 3.0347184e-07, 0.00135553, 232.23072781, 4.4060252e-07, 0.00186092, 265.91488444, 6.34979338e-07, 0.00255563, 303.66527709, 9.08780921e-07, 0.00351167, 345.84042051, 1.29217259e-06, 0.00482716, 392.81057213, 1.82595674e-06, 0.00663574, 444.95503676, 2.56502452e-06, 0.00912121, 502.65896662, 3.58284322e-06, 0.01254065, 566.30964378, 4.97730118e-06, 0.0172621, 636.29224142, 6.87837002e-06, 0.02382463, 712.98507019, 9.45826518e-06, 0.03303184, 796.75432742, 0.00001295, 0.04609141, 887.94837929, 0.00001764, 0.06481254, 986.89161935, 0.00002395, 0.09186755, 1093.87796045, 0.00003242, 0.13110992, 1209.16403126, 0.00004377, 0.18792081, 1332.96216285, 0.00005902, 0.26953291, 1465.43326394, 0.00007954, 0.38526828, 1606.67969676, 0.00010729, 0.5466472, 1756.7382768, 0.00014498, 0.76740202, 1915.57352952, 0.00019651, 1.06358703, 2083.07134522, 0.00026741, 1.45421886, 2259.03317838, 0.00036564, 1.96318886, 2443.17094032, 0.00050259, 2.6234934, 2635.10273319, 0.0006946, 3.48501062, 2834.34956926, 0.00096501, 4.62695524, 3040.33321162, 0.00134711, 6.17561973, 3252.37526058, 0.00188808, 8.32703362, 3469.69759509, 0.00265452, 11.3729761, 3691.42425933, 0.00373986, 15.72796342, 3916.58486232, 0.00527443, 21.9553048, 4144.11953295, 0.00743897, 30.79291666, 4372.8854449, 0.01048271, 43.18440517, 4601.66489546, 0.01474745, 60.32661892, 4829.17489113, 0.02069964, 83.74841602, 5054.07816002, 0.02897304, 115.43296091, 5274.99547934, 0.04042573, 157.98504791, 5490.5191744, 0.05621621, 214.82764097, 5699.22761618, 0.07790557, 290.3963241, 5899.70051719, 0.10759477, 390.30063771, 6090.53480154, 0.14810919, 521.45212851, 6270.36080551, 0.20324626, 692.22797136, 6437.85854949, 0.27810559, 912.83909206, 6591.77381288, 0.37952491, 1196.17952586, 6730.93373884, 0.51664738, 1559.51928777, 6854.26169792, 0.70164652, 2027.44995045, 6960.79114731, 0.95063099, 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197.68679332, 1657184.58052985, 4443.94453256, 226.91156566, 1734235.54691925, 4215.52632373, 258.69927939, 1747698.03680531, 3987.97484125, 292.98364531, 1694630.06174663, 3762.44761683, 329.62490861, 1579802.54040474, 3540.021994, 368.39441138, 1414987.26804181, 3321.68832026, 408.96099389, 1216898.75295298, 3108.34468091, 450.88150949, 1004298.42177957, 2900.79316354, 493.59495038, 794973.12918851, 2699.73761804, 536.42645118, 603276.6521061, 2505.78285297, 578.59845997, 438696.45434601, 2319.43517548, 619.25113233, 305575.64729172, 2141.10419099, 657.47106714, 203805.86150866, 1971.10571691, 692.33047351, 130108.52208616, 1809.66571206, 722.93067275, 79477.4937625, 1656.92505713, 748.44549138, 46440.77834305, 1512.945045, 768.17062695, 25950.62040493, 1377.71348382, 781.5619254, 13863.49191619, 1251.15115735, 788.27656983, 7078.88511422, 1133.11870644, 788.1732901, 3453.99391366, 1023.42351408, 781.34850203, 1610.07387294, 921.82673613, 768.10324261, 716.87944996, 828.05008738, 748.94914653, 304.81495148, 741.78311024, 724.56478996, 123.74702514, 662.68885363, 695.7116279, 47.95841126, 590.41092018, 663.36929537, 17.74000945, 524.57831467, 628.29563591, 0, 464.81160344, 591.44614496, 0, 410.72740327, 553.70939575, 0, 361.94252168, 515.76840044, 0, 318.07805157, 478.11871241, 0, 278.7642647, 441.27687267, 0, 243.63800016, 405.37066657, 0, 212.35173217, 371.20092364, 0, 184.57575619, 338.19830335, 0, 159.98660387, 307.05836193, 0, 138.28629587, 277.65440488, 0, 119.2111415, 249.81767463, 0, 102.48417889, 223.40944888, 0, 87.83198148, 198.65219107, 0, 0, 175.64614811, 0, 0, 154.30801511, 0, 0, 134.20480339, 0, 0, 115.90590215, 0, 0, 99.06947277, 0, 0, 83.86417027, 0, 0, 70.21045431, 0, 0, 58.47211449, 0, 0, 47.78640658, 0, 0, 38.68015927, 0, 0, 31.08724468, 0, 0, 24.57176017, 0, 0, 19.05428931, 0, 0, 14.62743547, 0, 0, 11.09310835, 0, 0, 8.12557309, 0, 0, 5.78101079, 0, 0, 4.19072833, 0, 0, 2.98005447, 0, 0, 2.05841601, 0, 0, 1.41472616, 0, 0, 0.87556236, 0, 0, 0.56659694, 0, 0, 0.27623448, 0, 0, 0.13726909, 0, 0, 0.07535991, 0, 0, 0.00128593, 0, 0, 6.14151393e-14]
@@ -141,6 +151,11 @@
           "type": "character",
           "attributes": {},
           "value": ["1/year"]
+        },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["average"]
         }
       },
       "value": [1620.54514448, 3661.04629032, 2275.82527489, 1719.21961641, 3814.15780294, 2379.51209696, 1829.29259114, 3966.98027609, 2486.1931826, 1954.47042106, 4120.12833196, 2597.19648495, 2099.41827813, 4274.59597231, 2714.30377717, 2269.88402805, 4431.81229397, 2839.81273406, 2472.97119376, 4593.75318514, 2976.66805246, 2717.76271737, 4763.18470552, 3128.75494023, 3016.55960122, 4944.13730734, 3301.47718144, 3387.03271994, 5142.72292206, 3502.75782539, 3855.60732856, 5368.41492685, 3744.61037537, 4462.45947529, 5635.93345396, 4045.45598228, 5268.68001827, 5967.94487815, 4433.44385342, 6366.52902481, 6398.92235621, 4951.20203081, 7894.2835747, 6980.73248674, 5662.71432733, 10057.89896382, 7790.78276934, 6663.3511507, 13162.34182064, 8943.80465335, 8094.37756417, 17655.62334944, 10608.41079716, 10163.34050572, 24187.65464356, 13029.22440558, 13171.31751002, 33683.21782699, 16554.31507917, 17546.69988534, 47422.61081102, 21665.51878975, 23882.5276523, 67114.14430287, 29005.69285575, 32970.05115546, 94929.9731724, 39392.18264225, 45815.31596154, 133463.22807928, 53800.69433859, 63619.3098504, 185555.60981256, 73300.45788892, 87698.13483213, 253948.77500362, 98923.13075919, 119321.59605289, 340738.59801697, 131457.57812784, 159460.52424619, 446663.19814219, 171182.14471759, 208457.13322897, 570328.37048315, 217573.38885887, 265666.39597558, 707549.55089012, 269058.63610953, 329151.37478533, 851037.95108271, 322897.94898103, 395537.89854769, 990647.41545724, 375276.94122691, 460128.84803171, 1114306.87833193, 421657.39401238, 517335.86651884, 1209596.65256261, 457369.96776551, 561409.1568167, 1265724.86703646, 478357.37940268, 587352.54203482, 1275489.94738354, 481912.33153107, 591832.06338668, 1236747.07412674, 467228.9258368, 573854.92393592, 1152975.08152589, 435615.82153971, 535031.94518009, 1032757.28889475, 390301.00812818, 479337.18655989, 888280.82881356, 335867.51171399, 412413.14163994, 733225.23567781, 277459.4506837, 340593.17049236, 580559.38963572, 219953.63438451, 269881.50816155, 440748.95181286, 167285.51903186, 205123.33150142, 320711.3575943, 122056.26643563, 149520.94427661, 223613.10023525, 85456.52682781, 104539.94006539, 149374.93780154, 57456.42112185, 70143.13363235, 95606.1241888, 37156.84814097, 45223.12927311, 58656.29975793, 23185.03508756, 28089.36457965, 34535.44992601, 14040.48049773, 16894.0749195, 19562.92181558, 8339.01387773, 9932.80762432, 10717.41072855, 4944.41392995, 5806.49006271, 5738.16004316, 3006.46599055, 3468.28624791, 3062.76702522, 1937.58941131, 2194.82622575, 1685.98499243, 1359.73855506, 1520.81397027, 1002.5541025, 1045.30888728, 1166.26522618, 670.22238186, 865.62155379, 973.06208835, 506.77497105, 752.18374209, 857.22714566, 420.78073739, 670.68927904, 776.94707081, 368.8581496, 604.83004153, 712.51119238, 327.09752245, 545.67000685, 653.34633925, 299.53010676, 494.75881605, 601.77244557, 273.34665494, 447.2566772, 552.18295719, 248.61057089, 403.11451774, 504.8142764, 225.39486001, 362.29780848, 459.9230287, 203.64309231, 324.61898718, 417.46272323, 183.53033047, 290.17322795, 377.92257889, 164.77398167, 258.51283087, 340.72132496, 147.52016629, 229.70620907, 306.27511586, 131.64441792, 203.50574034, 274.36368455, 117.0174714, 179.66936905, 244.74755871, 103.5165623, 157.9584528, 217.2055183, 91.14182445, 138.28287915, 191.80186385, 59.29457291, 71.09792507, 138.15396045, 52.09125251, 62.46069051, 121.37051478, 45.30481645, 54.32332652, 105.55839282, 39.12747897, 46.91631006, 91.16544594, 33.44384229, 40.10127195, 77.92280201, 28.31084093, 33.94648023, 65.96311611, 23.70162367, 28.41973862, 55.22382604, 19.73899851, 23.66830164, 45.99106943, 16.13172051, 19.34294826, 37.58625734, 13.05763633, 15.65692784, 30.42376535, 10.49442254, 12.58347317, 24.45158074, 8.29492728, 9.94613991, 19.32684559, 6.43234115, 7.71278191, 14.98709514, 4.93792519, 5.92088311, 11.50516628, 3.74480812, 4.49026065, 8.7252517, 2.74302848, 3.2890638, 6.39114558, 1.95155185, 2.34003351, 4.54703701, 1.41470479, 1.69632008, 3.29620503, 1.00600587, 1.20626436, 2.34395306, 0.69487945, 0.83320419, 1.61904105, 0.47758283, 0.57265187, 1.1127487, 0.29557207, 0.35440952, 0.68867099, 0.19127162, 0.22934671, 0.44565515, 0.09325115, 0.111814, 0.21727142, 0.04633926, 0.05556369, 0.1079686, 0.02543998, 0.03050413, 0.05927412, 0.0004341, 0.00052052, 0.00101145, 2.07325039e-14, 2.48595772e-14, 4.8305897e-14]
@@ -250,6 +265,11 @@
           "type": "character",
           "attributes": {},
           "value": ["1/year"]
+        },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["average"]
         }
       },
       "value": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5]
@@ -301,6 +321,11 @@
           "type": "character",
           "attributes": {},
           "value": ["1/year"]
+        },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["average"]
         }
       },
       "value": [1620.73220406, 3661.13276768, 2275.8428408, 1719.40667599, 3814.2442803, 2379.52966286, 1829.47965071, 3967.06675345, 2486.2107485, 1954.65748064, 4120.21480932, 2597.21405085, 2099.6053377, 4274.68244967, 2714.32134307, 2270.07108763, 4431.89877133, 2839.83029996, 2473.15825333, 4593.8396625, 2976.68561836, 2717.94977694, 4763.27118288, 3128.77250613, 3016.74666079, 4944.2237847, 3301.49474734, 3387.21977951, 5142.80939942, 3502.77539129, 3855.79438813, 5368.50140421, 3744.62794127, 4462.64653486, 5636.01993132, 4045.47354818, 5268.86707784, 5968.03135551, 4433.46141933, 6366.71608439, 6399.00883357, 4951.21959671, 7894.47063427, 6980.8189641, 5662.73189323, 10058.08602339, 7790.8692467, 6663.3687166, 13162.52888021, 8943.89113071, 8094.39513007, 17655.81040901, 10608.49727453, 10163.35807162, 24187.84170314, 13029.31088294, 13171.33507592, 33683.40488657, 16554.40155653, 17546.71745124, 47422.7978706, 21665.60526711, 23882.5452182, 67114.33136244, 29005.77933311, 32970.06872136, 94930.16023198, 39392.26911961, 45815.33352744, 133463.41513885, 53800.78081595, 63619.3274163, 185555.79687214, 73300.54436628, 87698.15239804, 253948.9620632, 98923.21723655, 119321.6136188, 340738.78507655, 131457.6646052, 159460.54181209, 446663.38520176, 171182.23119495, 208457.15079487, 570328.55754272, 217573.47533623, 265666.41354148, 707549.7379497, 269058.72258689, 329151.39235123, 851038.13814228, 322898.03545839, 395537.9161136, 990647.60251681, 375277.02770427, 460128.86559761, 1114307.06539151, 421657.48048974, 517335.88408474, 1209596.83962218, 457370.05424287, 561409.17438261, 1265725.05409603, 478357.46588004, 587352.55960072, 1275490.13444311, 481912.41800843, 591832.08095258, 1236747.26118632, 467229.01231416, 573854.94150182, 1152975.26858546, 435615.90801707, 535031.96274599, 1032757.47595433, 390301.09460554, 479337.20412579, 888281.01587313, 335867.59819135, 412413.15920584, 733225.42273738, 277459.53716106, 340593.18805826, 580559.57669529, 219953.72086187, 269881.52572745, 440749.13887244, 167285.60550922, 205123.34906732, 320711.54465387, 122056.35291299, 149520.96184251, 223613.28729482, 85456.61330517, 104539.95763129, 149375.12486111, 57456.50759921, 70143.15119825, 95606.31124837, 37156.93461833, 45223.14683901, 58656.4868175, 23185.12156492, 28089.38214555, 34535.63698559, 14040.56697509, 16894.0924854, 19563.10887516, 8339.10035509, 9932.82519022, 10717.59778813, 4944.50040731, 5806.50762861, 5738.34710273, 3006.55246791, 3468.30381381, 3062.9540848, 1937.67588867, 2194.84379165, 1686.17205201, 1359.82503242, 1520.83153617, 1003.24116207, 1045.39536464, 1166.28279208, 670.90944144, 865.70803115, 973.07965425, 507.46203062, 752.27021945, 857.24471156, 421.46779696, 670.7757564, 776.96463671, 369.54520917, 604.91651889, 712.52875828, 327.78458203, 545.75648421, 653.36390516, 300.21716633, 494.84529341, 601.79001147, 274.03371452, 447.34315456, 552.20052309, 249.29763046, 403.2009951, 504.8318423, 226.08191959, 362.38428584, 459.9405946, 204.33015188, 324.70546454, 417.48028913, 184.21739004, 290.25970531, 377.94014479, 165.46104124, 259.09930823, 340.73889086, 148.20722587, 230.29268643, 306.29268176, 132.3314775, 204.0922177, 274.38125045, 117.70453097, 180.25584641, 244.76512461, 104.20362187, 158.54493016, 217.2230842, 91.82888402, 138.86935651, 191.81942975, 59.98163249, 71.68440243, 138.17152635, 52.77831208, 63.04716787, 121.38808069, 45.99187602, 54.90980388, 105.57595872, 39.81453854, 47.50278742, 91.18301184, 34.13090186, 40.68774931, 77.94036791, 28.9979005, 34.53295759, 65.98068201, 24.38868325, 29.00621598, 55.24139194, 20.42605808, 24.254779, 46.00863533, 16.81878008, 19.92942562, 37.60382324, 13.7446959, 16.2434052, 30.94133125, 11.18148211, 13.16995053, 24.96914664, 8.98198685, 10.53261727, 19.84441149, 7.11940073, 8.29925927, 15.50466104, 5.62498476, 6.50736047, 12.02273218, 4.43186769, 5.07673801, 9.2428176, 3.43008805, 3.87554116, 6.90871148, 2.63861143, 2.92651087, 5.06460291, 2.10176436, 2.28279745, 3.81377093, 1.69306545, 1.79274172, 2.86151896, 1.38193902, 1.41968156, 2.13660695, 1.1646424, 1.15912923, 1.6303146, 0.98263164, 0.94088688, 1.20623689, 0.87833119, 0.81582407, 0.96322105, 0.78031072, 0.69829136, 0.73483732, 0.73339883, 0.64204105, 0.6255345, 0.71249955, 0.61698149, 0.57684002, 0.68749368, 0.58699788, 0.51857735, 0.68705957, 0.58647736, 0.5175659]
@@ -316,6 +341,11 @@
           "attributes": {},
           "value": [3, 100]
         },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["point"]
+        },
         "dimnames": {
           "type": "list",
           "attributes": {
@@ -367,6 +397,11 @@
           "attributes": {},
           "value": [3, 100]
         },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["average"]
+        },
         "dimnames": {
           "type": "list",
           "attributes": {
@@ -446,6 +481,11 @@
           "attributes": {},
           "value": [3, 100]
         },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["point"]
+        },
         "dimnames": {
           "type": "list",
           "attributes": {
@@ -497,6 +537,11 @@
           "attributes": {},
           "value": [3, 100]
         },
+        "representation": {
+          "type": "character",
+          "attributes": {},
+          "value": ["point"]
+        },
         "dimnames": {
           "type": "list",
           "attributes": {
diff --git a/tests/testthat/test-ArraySpeciesBySize.R b/tests/testthat/test-ArraySpeciesBySize.R
index 0fbe1977f..fdb9eb526 100644
--- a/tests/testthat/test-ArraySpeciesBySize.R
+++ b/tests/testthat/test-ArraySpeciesBySize.R
@@ -252,3 +252,60 @@ test_that("ArraySpeciesBySize value_name attribute", {
     mort <- getMort(NS_params_small)
     expect_identical(attr(mort, "value_name"), "Total mortality")
 })
+
+# Second-order plotting placement (#382) -----------------------------------
+
+test_that("producers tag the representation honestly", {
+    # Bin-averaged sinks are tagged "average"; flux/point quantities "point".
+    expect_identical(attr(getPredMort(NS_params_small), "representation"),
+                     "average")
+    expect_identical(attr(getMort(NS_params_small), "representation"),
+                     "average")
+    expect_identical(attr(getFMort(NS_params_small), "representation"),
+                     "average")
+    expect_identical(attr(getExtMort(NS_params_small), "representation"),
+                     "average")
+    expect_identical(attr(getEncounter(NS_params_small), "representation"),
+                     "point")
+    expect_identical(attr(getEGrowth(NS_params_small), "representation"),
+                     "point")
+})
+
+test_that("bin_midpoints are the geometric bin centres", {
+    p <- NS_params_small
+    beta <- p@w_full[2] / p@w_full[1]
+    expect_equal(bin_midpoints(p), p@w * sqrt(beta))
+    expect_equal(bin_midpoints(p, w_full = TRUE), p@w_full * sqrt(beta))
+})
+
+test_that("the size axis honours representation only under second_order_w", {
+    p <- NS_params_small
+    # Default first-order model: averaged quantities still plot at the nodes,
+    # so existing plots are unchanged.
+    expect_equal(get_ArraySpeciesBySize_w(getPredMort(p)), p@w)
+    expect_equal(get_ArraySpeciesBySize_w(getEncounter(p)), p@w)
+
+    second_order_w(p) <- c(bin_average = TRUE)
+    # Now averaged quantities move to the geometric bin centres ...
+    expect_equal(get_ArraySpeciesBySize_w(getPredMort(p)), bin_midpoints(p))
+    # ... while point-valued quantities stay on the nodes.
+    expect_equal(get_ArraySpeciesBySize_w(getEncounter(p)), p@w)
+})
+
+test_that("as.data.frame x-values follow the representation tag", {
+    p <- NS_params_small
+    second_order_w(p) <- c(bin_average = TRUE)
+    df_avg <- as.data.frame(getPredMort(p))
+    df_pt  <- as.data.frame(getEncounter(p))
+    expect_equal(sort(unique(df_avg$w)), sort(unname(bin_midpoints(p))))
+    expect_equal(sort(unique(df_pt$w)),  sort(unname(p@w)))
+})
+
+test_that("subsetting preserves the representation tag", {
+    p <- NS_params_small
+    second_order_w(p) <- c(bin_average = TRUE)
+    mort <- getPredMort(p)
+    sub <- mort[1:2, ]
+    expect_identical(attr(sub, "representation"), "average")
+    expect_equal(get_ArraySpeciesBySize_w(sub), bin_midpoints(p))
+})
diff --git a/tests/testthat/test-ArrayTimeBySpeciesBySize.R b/tests/testthat/test-ArrayTimeBySpeciesBySize.R
index 28a6c519e..a0711a1ea 100644
--- a/tests/testthat/test-ArrayTimeBySpeciesBySize.R
+++ b/tests/testthat/test-ArrayTimeBySpeciesBySize.R
@@ -250,3 +250,34 @@ test_that("animateSpectra is a backward-compatible alias for animate", {
     expect_s3_class(animateSpectra(fmort_small), "plotly")
     expect_s3_class(animateSpectra(NS_sim_small), "plotly")
 })
+
+# Second-order plotting placement (#382) -----------------------------------
+
+test_that("the time-series size axis honours representation under second_order_w", {
+    # getFMort.MizerSim tags its result "average".
+    expect_identical(attr(fmort_small, "representation"), "average")
+    p <- NS_sim_small@params
+    # Default first-order: size axis stays on the nodes (df unchanged). The
+    # stored dimnames are rounded to 3 sig figs, so compare with tolerance.
+    df_off <- as.data.frame(fmort_small)
+    expect_equal(sort(unique(df_off$w)), sort(unname(p@w)),
+                 tolerance = 1e-2)
+
+    # Switch the model to second order and recompute on the sim.
+    sim2 <- NS_sim_small
+    second_order_w(sim2@params) <- c(bin_average = TRUE)
+    fmort2 <- getFMort(sim2)
+    expect_equal(get_ArrayTimeBySpeciesBySize_w(fmort2), bin_midpoints(p))
+    df_on <- as.data.frame(fmort2)
+    expect_equal(sort(unique(df_on$w)), sort(unname(bin_midpoints(p))))
+})
+
+test_that("a time slice keeps the representation tag", {
+    sim2 <- NS_sim_small
+    second_order_w(sim2@params) <- c(bin_average = TRUE)
+    fmort2 <- getFMort(sim2)
+    slice <- ArrayTimeBySpeciesBySize_slice(fmort2)
+    expect_s3_class(slice, "ArraySpeciesBySize")
+    expect_identical(attr(slice, "representation"), "average")
+    expect_equal(get_ArraySpeciesBySize_w(slice), bin_midpoints(sim2@params))
+})
diff --git a/tests/testthat/test-project_methods.R b/tests/testthat/test-project_methods.R
index e6bc1c26c..2fcb395e8 100644
--- a/tests/testthat/test-project_methods.R
+++ b/tests/testthat/test-project_methods.R
@@ -213,7 +213,8 @@ test_that("species-size rate getters work for MizerSim", {
 test_that("getCriticalFeedingLevel matches metab over intake_max times alpha", {
     expected <- params@metab / params@intake_max / params@species_params$alpha
     expect_equal(getCriticalFeedingLevel(params), expected,
-                 ignore_attr = c("value_name", "units", "class", "params"))
+                 ignore_attr = c("value_name", "units", "class", "params",
+                                 "representation"))
 })
 
 # getPredRate -------------------------------------------------------------
diff --git a/vignettes/numerical_details.qmd b/vignettes/numerical_details.qmd
index 9caf3e7bc..56599edb5 100644
--- a/vignettes/numerical_details.qmd
+++ b/vignettes/numerical_details.qmd
@@ -106,6 +106,8 @@ In summary:
 
 The default scheme uses point values throughout, which is consistent. Replacing the bin-integrated quantities by their bin averages raises the order of those source and sink terms; the spatial order of the overall transport step is separately set by the upwind advective flux, as discussed in the Order of Accuracy section below.
 
+**Plotting follows the same distinction.** A bin average $N_j$ does not live at the bin boundary $w_j$ but at the geometric bin centre $w^*_j=\sqrt{w_j\,w_{j+1}}=w_j\sqrt\beta$ (the log-midpoint, exact for the community spectrum $N\propto w^{-2}$). So under second-order bin-averaging mizer draws bin-averaged quantities (the abundance and the mortality/reproduction sinks) at $w^*_j$ — a uniform half-bin shift to the right on the log axis — while point-valued quantities (the encounter and growth-type rates) stay on the nodes $w_j$. The size-resolved array classes carry a `representation` tag recording which a quantity is, and the shift is applied only when `second_order_w[["bin_average"]]` is set, so default plots are unchanged.
+
 ## Semi-Implicit Time Discretisation
 
 With the diffusion term, an explicit time discretisation would require a very small time step for stability ($\Delta t \sim \Delta w^2$). Therefore, we use a semi-implicit scheme where the densities $N$ are evaluated at time $t+1$, but the rates ($g$, $\mu$, $d$) are evaluated at time $t$. Using a fully implicit scheme would require solving a nonlinear system at each time step, which is more computationally expensive.