Chapter 12 Seaward Movement

12.1 Overview

This function simulates seaward movement for fish moving downstream through riverine and estuarine systems. Agents respond to hydrodynamic conditions by adjusting swimming difficulty, swimming speed, and movement along a least cost pathway toward a downstream movement patch.

This submodel represents downstream movement following the same mechanistic framework as landward movement, where movement emerges from the interaction between hydrodynamic forcing, body size, and energetic state (Brown et al. 2004; Gibson et al. 2001a). In this case, movement occurs in the seaward direction, requiring a change in sign within velocity-based scaling terms to reflect alignment with downstream flow. As in landward movement, agents follow a least-cost pathway based on local hydrodynamic resistance, allowing downstream trajectories, travel time, and energy expenditure to emerge dynamically from the interaction between organismal capacity and the flow field.

12.2 Purpose

The purpose of this submodel is to represent downstream movement under realistic hydrodynamic resistance. Agents evaluate flow velocity, movement difficulty, energetic condition, and travel costs before advancing along a least cost route toward a movement destination.

12.3 Entities, State Variables, and Scales

12.3.1 Spatial and Temporal Scales

Spatial Unit: Patch (3 m x 3 m resolution)
Temporal Unit: 5 minute time steps (tick)

12.3.2 Global Variables

Variable	Definition
max-seaward-velocity	Maximum observed seaward velocity in the domain.
max-landward-velocity	Maximum observed landward velocity in the domain.

12.3.3 Patch Variables

Variable	Definition
velocity	Depth averaged hydrodynamic velocity.
depth	Water depth at the patch.
patch-terrain	Indicates whether the patch is land or water.
cost-to-sea	Patch-level travel cost for seaward movement.
visits-by-alewife	Number of times the agent visits the patch.
ticks-spent-alewife	Time spent on the patch.

12.3.4 Agent Variables

Variable	Definition
weight	Body mass of the fish.
speed	Current swimming speed.
prev-speed	Previous time step swimming speed.
max-speed	Maximum sustained speed.
min-speed	Minimum allowable speed.
swim-efficiency	Swimming acceleration and energy scaling.
difficulty-factor	Hydrodynamic difficulty.
planned-path	Sequence of patches in least cost route.
migration-patch	Destination patch.
trail	Record of visited patches.

12.4 Process Overview and Scheduling

Calculate swimming difficulty using normalized velocity and body size.
Update swimming speed using energy reserves, difficulty, and flow velocity.
Calculate cost-to-sea and construct least cost path.
Move along the first part of the path based on swimming speed.
Record patch visits and time spent.

12.5 Design Concepts

Basic Principles: Hydrodynamic forcing, body size, and energy availability determine downstream movement.

Emergence: Route choice, travel duration, and energetic depletion emerge from repeated state updates.

Objectives: Agents attempt to reach the seaward migration patch following a hydrodynamically efficient route.

Sensing: Agents sense local velocity, slope, difficulty, and patch cost gradients.

Observation: Movement trajectories, energy decline, and convergence routes can be monitored.

12.6 Initialization

Variable	Initial Value	Justification
velocity	Hydrodynamic input	Required environmental forcing.
weight	Species-specific	Required for difficulty scaling.
max-speed	Species-specific	Physiological constraint.
min-speed	Species-specific	Prevents collapse to zero speed.
swim-efficiency	Species or constant	Controls acceleration and deceleration.
difficulty-factor	1	Neutral start value.
prev-speed	Starting speed	Smooth initial movement.

12.7 Submodels

12.7.1 Swimming Difficulty

Velocity is normalized between minimum and maximum observed values:

\[ V_{normalized} = \frac{V_{\text{patch}} - V_{\min}} {V_{\max} - V_{\min}} \hspace{2cm} \text{(12.1)} \]

Where:

\(V_{normalized}\) is the normalized patch velocity (dimensionless).
\(V_{patch}\) is the depth-averaged along-channel water velocity at the patch (m s⁻¹).
\(V_{min}\) is the minimum observed velocity (m s⁻¹).
\(V_{max}\) is the maximum observed velocity (m s⁻¹).

Difficulty scales with body size (Brown et al. 2004; Gibson et al. 2001a):

\[ Df_{\text{raw}} = \left( \frac{V_{normalized}} {A_{\text{mass}} / M_{\max}} \right)^k \hspace{2cm} \text{(12.2)} \]

Where:

\(Df_{raw}\) is the raw hydrodynamic difficulty prior to mapping (dimensionless).
\(V_{normalized}\) is the normalized patch velocity (dimensionless).
\(A_{mass}\) is an agent’s body mass used for allometric and energetic scaling (g).
\(M_{max}\) is the maximum body mass across the population (g).
\(k\) is the energy scaling exponent (dimensionless).

Difficulty is mapped onto the one-to-ten range:

\[ D_f = 1 + 9 \cdot Df_{\text{raw}} \hspace{2cm} \text{(12.3)} \]

Where:

\(D_f\) is the mapped swimming difficulty index (dimensionless).
\(Df_{raw}\) is the raw hydrodynamic difficulty prior to mapping (dimensionless).

and constrained:

\[ D_f \in [1, 10] \hspace{2cm} \text{(12.4)} \]

Where:

\(D_f\) is the mapped swimming difficulty index (dimensionless, range 1-10).

12.7.2 Swimming Speed

Energy factor:

\[ E_{\text{factor}} = \frac{E_{\text{agent}}}{100} \hspace{2cm} \text{(12.5)} \]

Where:

\(E_{factor}\) is the normalized energy level of the agent (dimensionless).
\(E_{agent}\) is the total internal energy available to the agent (energy units).

Velocity impact from hydrodynamics is:

\[ V_{\text{impact}} = k \cdot V_{\text{patch}} \cdot 300 \hspace{2cm} \text{(12.6)} \]

Where:

\(V_{impact}\) is the flow contribution to desired swimming speed (m tick⁻¹).
\(k\) is the energy scaling exponent used in velocity, difficulty, and osmoregulation terms (dimensionless).
\(V_{patch}\) is the depth-Averaged along-channel water velocity at the patch (m s⁻¹).
\(300\) is the timestep conversion from seconds to one tick (s/tick, e.g., 5 minutes).

The desired swimming speed is:

\[ V_{\text{desired}} = \frac{V_{\max} \cdot E_{\text{factor}}}{D_f} + V_{\text{impact}} \hspace{2cm} \text{(12.7)} \]

Where:

\(V_{desired}\) is the desired swimming speed before constraints (m tick⁻¹).
\(V_{max}\) is the maximum observed velocity (m tick⁻¹).
\(E_{factor}\) is the normalized energy level of the agent (dimensionless).
\(D_f\) is the mapped swimming difficulty index (dimensionless).
\(V_{impact}\) is the flow contribution to desired swimming speed (m tick⁻¹).

Speed is constrained between a minimum and maximum:

\[ V_{\text{desired}} = \min(V_{\max}, \max(V_{\min}, V_{\text{desired}})) \hspace{2cm} \text{(12.8)} \]

Where:

\(V_{desired}\) is the desired swimming speed before constraints (m tick⁻¹).
\(V_{max}\) is the maximum observed velocity (m tick⁻¹).
\(V_{min}\) is the minimum observed velocity (m tick⁻¹).

Acceleration smoothing:

\[ \Delta V_{\max} = 0.5 \cdot swim_{Eff} \hspace{2cm} \text{(12.9)} \]

Where:

\(\Delta V_{max}\) is the maximum allowable speed change per tick (m tick⁻¹).
\(swim_{Eff}\) is the efficiency multiplier applied to swimming energetics (%).

If \(|V_{\text{desired}} - V_{\text{prev}}| > \Delta V_{\max}\),where \(V_{prev}\) is the previous speed then speed moves gradually toward the desired value. Otherwise speed is set directly.

12.7.3 Path-Based Movement

Travel distance:

\[ D_{travel} = \frac{A_{\text{speed}}}{3} \hspace{2cm} \text{(12.10)} \]

Where:

\(D_{travel}\) is the distance traveled in the current tick (patch units), where 1 patch = 3 m.
\(A_{speed}\) is the actual swimming speed used for movement (m tick⁻¹).

Movement follows:

compute cost-to-sea
build path using decreasing cost
traverse first \(\lceil D_{travel} \rceil\) patches
record visitation and time spent
append to the movement trail

12.7.4 Travel Cost

Raw cost:

\[ \text{cost}_{\text{raw}} = |V_{\text{patch}}| + 0.5 \cdot \max(0, \Delta d) \hspace{2cm} \text{(12.11)} \]

Where:

\(cost_{raw}\) is the base travel cost including velocity and slope (dimensionless).
\(V_{patch}\) is the depth-averaged along-channel water velocity at the patch (m s⁻¹).
\(\Delta d\) is the change in depth between patches (m).

Difficulty scaling:

\[ \text{cost} = \text{cost}_{\text{raw}} \cdot D_f \hspace{2cm} \text{(12.12)} \]

Where:

\(cost\) is the difficulty-scaled travel cost per patch (energy units).
\(cost_{raw}\) is the base travel cost including velocity and slope (dimensionless).
\(D_f\) is the mapped swimming difficulty index (dimensionless).

This produces a cost-to-sea field representing hydrodynamic resistance (dimensionless), which is used for route selection.

12.7.5 Swimming Energy

Let \(\beta = 0.75\).

Energy multiplier:

\[ M_{swim} = \frac{D_f}{swim_{Eff}} \hspace{2cm} \text{(12.13)} \]

Where:

\(M_{swim}\) is the swimming energy multiplier (dimensionless).
\(D_f\) is the mapped swimming difficulty index (dimensionless).
\(swim_{Eff}\) is the efficiency multiplier applied to swimming energetics (%).

Energy cost:

\[ E_{\text{swim}} = swim_{base} \cdot M_{swim}^{\beta} \hspace{2cm} \text{(12.14)} \]

Where:

\(E_{swim}\) is the energy cost of swimming for the tick (energy units).
\(swim_{base}\) is the baseline energetic cost of swimming ((energy units) patch⁻¹).
\(M_{swim}\) is the swimming energy multiplier (dimensionless).
\(\beta\) is the exponent controlling swimming cost scaling (dimensionless).

Energy update:

\[ E_{\text{agent}} = \max \left(0,\, E_{\text{agent}} - E_{\text{swim}} \right) \hspace{2cm} \text{(12.15)} \]

Where:

\(E_{factor}\) is the normalized energy level of the agent (energy units).
\(E_{agent}\) is the total internal energy available to the agent (energy units).

12.8 NetLogo Implementation


;; ===================================================================
;;  SEAWARD MOVEMENT + HYDRODYNAMIC SWIMMING SUBMODEL DECLARATIONS
;; ===================================================================

globals [

  ;; Velocity normalization bounds
  max-seaward-velocity
  max-landward-velocity
]

patches-own [

  ;; Environmental attributes
  velocity
  depth
  patch-terrain   ;; "water" or "land"

  ;; Least-cost routing fields
  cost-to-home
  cost-to-sea
  cost-to-prey

  ;; Tracking fish patch-level behavior
  visits-by-fish
  ticks-spent-fish
]

turtles-own [

  ;; Body condition
  weight

  ;; Swimming behavior
  speed
  prev-speed
  max-speed
  min-speed
  swim-efficiency

  ;; Hydrodynamic resistance
  difficulty-factor

  ;; Energetics
  E-swim
  swim-base

  ;; Movement targets
  home-patch
  migration-patch

  ;; Movement memory
  planned-path
  trail
  previous-patch
  previous-x
  previous-y
]

;; ===================================================================
;;  SEAWARD MOVEMENT + HYDRODYNAMIC SWIMMING SUBMODEL
;; ===================================================================

;; -------------------------------------------------------------------
;;  Swimming Difficulty (Seaward)
;; -------------------------------------------------------------------
to calculate-difficulty-seaward
  
  let M-max max [weight] of breed
  let M-agent weight
  let V-max max-landward-velocity
  let V-min max-seaward-velocity
  let k 0.75

  let normalized-velocity (velocity - V-min) / (V-max - V-min)

  let Df-raw (normalized-velocity / (M-agent / M-max)) ^ k

  set difficulty-factor (1 + 9 * Df-raw)
  set difficulty-factor max list 1 (min list 10 difficulty-factor)
end


;; -------------------------------------------------------------------
;;  Swimming Speed (Seaward)
;; -------------------------------------------------------------------
to calculate-swimming-speed-seaward
  
  let k 0.75
  let energy-factor (energy / 100)

  let velocity-impact (k * velocity * 300)

  let desired-speed 
    (max-speed * energy-factor / difficulty-factor)
      + velocity-impact

  set desired-speed 
      min list max-speed (max list min-speed desired-speed)

  let max-speed-change 0.5
  set max-speed-change (max-speed-change * swim-efficiency)

  if abs(desired-speed - prev-speed) > max-speed-change [
    if desired-speed > prev-speed [
      set speed prev-speed + max-speed-change
    ]
    if desired-speed < prev-speed [
      set speed prev-speed - max-speed-change
    ]
  ]

  if abs(desired-speed - prev-speed) <= max-speed-change [
    set speed desired-speed
  ]

  set prev-speed speed
end


;; -------------------------------------------------------------------
;;  Cost-to-Sea (Least Cost Routing)
;; -------------------------------------------------------------------
to calculate-cost-to-sea [sea-patch]

  ask migration-patch [ set cost-to-sea 0 ]
  ask patch-here     [ set cost-to-sea 1e9 ]

  let frontier (list migration-patch)
  let df difficulty-factor

  while [not empty? frontier] [

    let current first frontier
    set frontier but-first frontier

    ask current [

      let current-cost cost-to-sea

      ask neighbors4 with [patch-terrain = "water"] [

        let travel-cost compute-travel-cost current self df
        let new-cost current-cost + travel-cost

        if new-cost < cost-to-sea [
          set cost-to-sea new-cost
          set frontier lput self frontier
        ]
      ]
    ]
  ]
end


;; -------------------------------------------------------------------
;;  Build Path to Sea
;; -------------------------------------------------------------------
to-report build-path-sea [current-patch sea-patch]

  let path (list current-patch)
  let cur current-patch
  let max-steps 2000

  while [cur != migration-patch and max-steps > 0] [

    set max-steps max-steps - 1

    let candidates [neighbors] of cur
    set candidates candidates with [
      patch-terrain = "water" and 
      cost-to-sea < [cost-to-sea] of cur
    ]

    if not any? candidates [
      report path
    ]

    let next-patch min-one-of candidates [cost-to-sea]
    set path lput next-patch path
    set cur next-patch
  ]

  report path
end


;; -------------------------------------------------------------------
;;  Seaward Movement – Movement Execution
;; -------------------------------------------------------------------
to migrate-seaward
  
  ;; Schooling logic (excluded in documentation)
  find-schoolmates
  if any? schoolmates [
    find-nearest-neighbor
    ifelse distance nearest-neighbor < minimum-separation [
      separate
    ][
      cohere
      align
    ]
    adjust-speed
  ]

  ;; Path-based movement
  let travel-distance (speed / 3)
  let path build-path-sea patch-here migration-patch
  set planned-path path

  if not empty? path [

    let steps-to-move ceiling travel-distance
    let move-patches sublist path 0 (min list steps-to-move length path)

    let old-patch patch-here

    foreach move-patches [ p ->

      if p != old-patch [
        ask p [ set visits-by-alewife visits-by-alewife + 1 ]
        set old-patch p
      ]

      face p
      move-to p

      ask p [
        set ticks-spent-alewife ticks-spent-alewife + 1
      ]
    ]
  ]

  if not member? patch-here trail [
    set trail lput patch-here trail
  ]
end


;; -------------------------------------------------------------------
;;  Universal Swimming Energy Cost
;; -------------------------------------------------------------------
to calculate-swim-energy
  
  let beta 0.75
  let energy-multiplier (difficulty-factor * (1 / swim-efficiency))

  set E-swim (swim-base * (energy-multiplier ^ beta))
  set energy max list 0 (energy - E-swim)
end

References

Brown, James H., James F. Gillooly, Andrew P. Allen, Van M. Savage, and Geoffrey B. West. 2004. “TOWARD a METABOLIC THEORY OF ECOLOGY.” Ecology 85 (7): 1771–89. https://doi.org/10.1890/03-9000.

Gibson, R. N., M. Barnes, and R. J. A. Atkinson. 2001a. “Selective Tidal-Stream Transport of Marine Animals.” Oceanography and Marine Biology: An Annual Review 39: 305–53.