Author: Hans Ttito
Date: 2026-05-04

# Temperature Sensitivity and Climate Change Scenarios

#### Hans Ttito

#### 2026-05-04

## Overview

Temperature is the primary environmental driver in fish bioenergetics. Because metabolic rates scale non-linearly with temperature, even modest warming can have disproportionate effects on growth and consumption. This vignette uses `fb4package` to:

1. Quantify how final weight responds to temperature shifts (sensitivity analysis)
2. Project growth outcomes under +1 °C, +2 °C, and +4 °C warming scenarios
3. Identify thermal performance windows using the sensitivity plot
4. Compare sensitivity across two ecologically contrasting species

## 1. Base model — juvenile Chinook salmon

We use 120-day summer conditions as the baseline.

    # Species parameters (Stewart & Ibarra 1991)
    sp_params_chinook <- list(
      consumption = list(
        CEQ = 2, CA = 0.303, CB = -0.275,
        CQ = 3.0, CTO = 15.0, CTM = 20.0,
        CTL = 24.0, CK1 = 0.1, CK4 = 0.13
      ),
      respiration = list(
        REQ = 1, RA = 0.00264, RB = -0.217,
        RQ = 0.06818, RTO = 0.0234, RTM = 0.0,
        RTL = 25.0, RK1 = 1.0, RK4 = 0.13, RK5 = 0.0
      ),
      activity  = list(ACT = 1.0, BACT = 0.0),
      sda       = list(SDA = 0.172),
      egestion  = list(EGEQ = 1, FA = 0.212, FB = -0.222, FG = 0.631),
      excretion = list(EXEQ = 1, UA = 0.0314, UB = 0.58, UG = -0.299),
      predator  = list(
        PREDEDEQ = 2,
        Alpha1 = 4500, Beta1 = 6.0,
        Alpha2 = 5500, Beta2 = 2.0,
        Cutoff = 250
      )
    )

    days <- 1:120

    # Baseline temperature: cool Pacific NW summer
    temp_base <- data.frame(
      Day         = days,
      Temperature = round(7 + 6 * sin(pi * (days - 20) / 120), 2)
    )

    # Diet
    diet_props  <- data.frame(Day = days, Alewife = 0.65, Shrimp = 0.35)
    prey_energy <- data.frame(Day = days, Alewife = 4900, Shrimp = 3200)

    # Build baseline object
    make_bio <- function(temp_df, sp_params, initial_weight = 5) {
      bio <- Bioenergetic(
        species_params     = sp_params,
        species_info       = list(scientific_name = "Oncorhynchus tshawytscha",
                                   common_name = "Chinook salmon",
                                   life_stage  = "juvenile"),
        environmental_data = list(temperature = temp_df),
        diet_data          = list(
          proportions = diet_props,
          prey_names  = c("Alewife", "Shrimp"),
          energies    = prey_energy
        ),
        simulation_settings = list(initial_weight = initial_weight, duration = 120)
      )
      bio
    }

    bio_base <- make_bio(temp_base, sp_params_chinook)

    cat(sprintf("Baseline temperature range : %.1f – %.1f °C\n",
                min(temp_base$Temperature), max(temp_base$Temperature)))
    #> Baseline temperature range : 4.1 – 13.0 °C

## 2. Manual sensitivity analysis across temperature offsets

We run binary search at each temperature offset and record the p required to achieve 40 g after 120 days (our target growth endpoint).

    offsets   <- c(-3, -2, -1, 0, 1, 2, 3, 4)
    target_wt <- 40

    results_sens <- lapply(offsets, function(delta) {
      temp_shifted <- temp_base
      temp_shifted$Temperature <- pmax(1, temp_base$Temperature + delta)

      bio_s <- make_bio(temp_shifted, sp_params_chinook)
      res   <- tryCatch(
        run_fb4(bio_s,
                fit_to    = "Weight",
                fit_value = target_wt,
                strategy  = "binary_search",
                verbose   = FALSE),
        error = function(e) NULL
      )

      if (is.null(res)) {
        return(data.frame(offset = delta, p_value = NA, final_weight = NA))
      }
      data.frame(
        offset       = delta,
        p_value      = round(res$summary$p_value,       4),
        final_weight = round(res$summary$final_weight,   1)
      )
    })

    sens_df <- do.call(rbind, results_sens)

    knitr::kable(
      sens_df,
      caption    = paste0("p-value required to achieve ", target_wt,
                          " g across temperature offsets"),
      col.names  = c("Temp. offset (°C)", "Estimated p", "Final weight (g)")
    )

p-value required to achieve 40 g across temperature offsets
Temp. offset (°C) | Estimated p | Final weight (g)
--- | --- | ---
-3 | 0.5100 | 40
-2 | 0.4705 | 40
-1 | 0.4381 | 40
0 | 0.4125 | 40
1 | 0.3936 | 40
2 | 0.3821 | 40
3 | 0.3794 | 40
4 | 0.3887 | 40

    plot(sens_df$offset, sens_df$p_value,
         type = "b", pch = 19, lwd = 2, col = "steelblue",
         xlab = "Temperature offset (°C)",
         ylab = "Estimated p-value",
         main = "Feeding level required to achieve target weight")
    abline(v = 0, lty = 2, col = "gray50")
    grid()

Estimated p-value as a function of temperature offset.

Interpretation: At warmer temperatures, Chinook must feed at a higher proportion of maximum ration to achieve the same growth target, because respiration costs increase faster than maximum consumption near the thermal optimum. Above ~+3 °C, the required p approaches or exceeds 1, indicating that growth to 40 g may become impossible at those temperatures.

## 3. Built-in sensitivity plot

`plot(bio, type = "sensitivity")` runs the same analysis internally and produces a formatted figure directly.

    plot(bio_base,
         type      = "sensitivity",
         temp_offsets = c(-3, -2, -1, 0, 1, 2, 3, 4),
         fit_to    = "Weight",
         fit_value = 40,
         colors    = "blue")
    #> === GROWTH SENSITIVITY ANALYSIS ===
    #> Mode: SEQUENTIAL
    #> Temperature range: 8 - 18 °C ( 6 values)
    #> P-value range: 0.3 - 1 ( 8 values)
    #> Total combinations: 48
    #>
    #> Processing temperature: 8 °C
    #> Processing temperature: 10 °C
    #> Processing temperature: 12 °C
    #> Processing temperature: 14 °C
    #> Processing temperature: 16 °C
    #> Processing temperature: 18 °C
    #>
    #> === ANALYSIS COMPLETE ===
    #> Successful simulations: 48 / 48 ( 100 %)
    #> Maximum growth rate: 1.768 %/d
    #> Optimal conditions: 14 °C, 100 % feeding (p = 1.00 )

Built-in temperature sensitivity plot. Each point shows the p required to achieve the target weight at a given temperature offset.

## 4. Climate change scenarios — projected growth

Rather than fixing a target weight, here we fix p = 0.65 (estimated from field data) and ask: how does final weight change under warming scenarios?

    p_fixed   <- 0.65
    scenarios <- c("Baseline" = 0, "+1 °C" = 1, "+2 °C" = 2, "+4 °C" = 4)

    scenario_results <- lapply(names(scenarios), function(sc) {
      delta <- scenarios[[sc]]
      temp_s <- temp_base
      temp_s$Temperature <- pmax(1, temp_base$Temperature + delta)

      bio_s <- make_bio(temp_s, sp_params_chinook)
      res   <- run_fb4(bio_s,
                       fit_to    = "p_value",
                       fit_value = p_fixed,
                       strategy  = "direct_p_value",
                       verbose   = FALSE)

      if (is.null(res)) return(NULL)

      data.frame(
        Scenario    = sc,
        Delta_T     = delta,
        Final_wt_g  = round(res$summary$final_weight,       1),
        Consumption = round(res$summary$total_consumption_g, 1)
      )
    })

    scenario_df <- do.call(rbind, scenario_results)

    knitr::kable(
      scenario_df,
      caption   = paste0("Projected outcomes at p = ", p_fixed,
                         " under four temperature scenarios"),
      col.names = c("Scenario", "Δ T (°C)", "Final weight (g)", "Total consumption (g)")
    )

Projected outcomes at p = 0.65 under four temperature scenarios
Scenario | Δ T (°C) | Final weight (g) | Total consumption (g)
--- | --- | --- | ---
Baseline | 0 | 111.1 | 264.1
+1 °C | 1 | 125.8 | 306.8
+2 °C | 2 | 137.4 | 343.6
+4 °C | 4 | 138.3 | 362.4

    bar_cols <- c("steelblue", "gold", "darkorange", "firebrick")
    bp <- barplot(scenario_df$Final_wt_g,
                  names.arg = scenario_df$Scenario,
                  col       = bar_cols,
                  ylab      = "Final weight (g)",
                  main      = paste("Growth at p =", p_fixed,
                                    "under warming scenarios"),
                  ylim      = c(0, max(scenario_df$Final_wt_g) * 1.25),
                  border    = NA)

    text(bp, scenario_df$Final_wt_g + 0.5,
         labels = paste0(scenario_df$Final_wt_g, " g"),
         cex = 0.85, font = 2)
    grid(nx = NA, ny = NULL, col = "gray85")

Projected final weight under climate change scenarios (p fixed at 0.65).

## 5. Thermal performance window

The metabolism–consumption balance defines a thermal performance window: the temperature range where the fish can maintain positive growth (p < 1). We can derive it by finding the temperature at which p = 1 for our target weight.

    # Sweep a fine temperature grid and record required p
    temp_grid   <- seq(2, 24, by = 1)
    p_at_target <- sapply(temp_grid, function(t_const) {
      temp_const <- data.frame(Day = days, Temperature = t_const)
      bio_t <- make_bio(temp_const, sp_params_chinook)
      tryCatch({
        res <- run_fb4(bio_t,
                       fit_to    = "Weight",
                       fit_value = target_wt,
                       strategy  = "binary_search",
                       verbose   = FALSE)
        res$summary$p_value
      }, error = function(e) NA_real_)
    })

    perf_df <- data.frame(Temperature = temp_grid, p_required = p_at_target)
    feasible <- perf_df[!is.na(perf_df$p_required) & perf_df$p_required <= 1, ]

    cat(sprintf("Feasible temperature range for %.0f g in %d days : %.0f – %.0f °C\n",
                target_wt, max(days),
                min(feasible$Temperature), max(feasible$Temperature)))
    #> Feasible temperature range for 40 g in 120 days : 2 – 19 °C

    plot(perf_df$Temperature, perf_df$p_required,
         type = "l", lwd = 2, col = "steelblue",
         xlab = "Constant water temperature (°C)",
         ylab = "p required to achieve 40 g",
         main = "Thermal performance window — juvenile Chinook",
         ylim = c(0, max(perf_df$p_required, na.rm = TRUE) * 1.1))

    # Shade feasible zone
    polygon(c(min(feasible$Temperature), feasible$Temperature,
              max(feasible$Temperature)),
            c(0, feasible$p_required, 0),
            col = adjustcolor("steelblue", alpha.f = 0.2),
            border = NA)

    abline(h = 1, lty = 2, col = "firebrick", lwd = 1.5)
    text(max(temp_grid) - 2, 1.02, "p = 1 (maximum ration)", col = "firebrick", cex = 0.8)
    grid()

Thermal performance window: p required to achieve 40 g over 120 days at constant temperatures. The shaded region shows the feasible window (p ≤ 1).

## 6. Management implications

These analyses have direct relevance for fish management under climate change:

- Habitat quality: The thermal performance window defines suitable temperature ranges for stocking or conservation planning.
- Consumption modelling: Holding p fixed while warming temperatures leads to lower growth at very warm temperatures, as respiration increases faster than consumption.
- Bioenergetics-based monitoring: Changes in condition factor (K) in monitoring data can be translated into shifts in p using this framework, providing a mechanistic link between climate variability and fish condition.

## References

Deslauriers D, Chipps SR, Breck JE, Rice JA, Madenjian CP (2017). Fish Bioenergetics 4.0: An R-Based Modeling Application. Fisheries 42(11):586–596. https://doi.org/10.1080/03632415.2017.1377558

Kitchell JF, Stewart DJ, Weininger D (1977). Applications of a bioenergetics model to yellow perch (Perca flavescens) and walleye (Stizostedion vitreum vitreum). Journal of the Fisheries Research Board of Canada 34(10):1922–1935. https://doi.org/10.1139/f77-258

Parmesan C, Yohe G (2003). A globally coherent fingerprint of climate change impacts across natural systems. Nature 421:37–42. https://doi.org/10.1038/nature01286