An introduction to some individual-based methods and to the exercises Wolf M. Mooij Netherlands...
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Transcript of An introduction to some individual-based methods and to the exercises Wolf M. Mooij Netherlands...
An introduction to some individual-based methods and to the exercises
Wolf M. Mooij
Netherlands Institute of Ecology
Nt+1 = r Nt (1 - Nt)
The logistic difference equation
Example of a deterministic, non-spatially explicit, unstructured, single species, population dynamical model based on
difference equations
Problem:
Given the research question, how to find the right model from the many models currently
around in ecology
Need for a classification of models
Disclaimer: the one presented here is just one of many possible classifications
How many species do we want to take into account?
• Single species
• Prey predator
• Tritrophic
• Multispecies
• Foodweb
Which part of the life-cycle do we want to take into account?
• Cohort-dynamics (only mortality)
• Population-dynamics (incl. reproduction)
Do we explicitly want to take space into account?
• Spatially implicit
• Spatially explicit -> at what scale?
How do we want to deal with variation between individuals?
• Unstructured model
• Individual-based distribution model (matrix model, physiologically structured model)
• Individual-based configuration model (individual-by-individual model)
Do we want to take stochastic processes into account?
• Deterministic model
• Stochastic model
Which mathematical format are we going to use?
• Difference equations (including matrices) -> at what time scale
• Differential equations (ordinary and partial)
• Logical if-then rules
Nt+1 = r Nt (1 - Nt)
The logistic model implemented as difference equation
• single species• population dynamical• spatially implicit• unstructured• deterministic• based on difference equations
dN/dt = r N (1 – N/K)
The logistic model implemented as differential equation
• single species• population dynamical• spatially implicit• unstructured• deterministic• based on differential equations
The big confusion between differential and difference equations: Euler integration
Differential equation: dN/dt = r Nt (1 – Nt/K)
Euler approximation: Nt+t = Nt + r Nt (1 – Nt/K) t
Difference equation: Nt+1 = Nt + r Nt (1 – Nt/K)
The latter two equations represent different models, especially when rates (r) are high!
dV/dt = r V (1 – N/K) – a V P / (H + V)
dP/dt = a c V P / (H + V) – m P
The Rosenzweig-MacArthur prey-predator model
• prey-predator• population dynamical• spatially implicit• unstructured• deterministic• based on differential equations
Tritrophic, multispecies, and foodweb extensions of these lotka-volterra type of models
• tritrophic, multispecies, foodweb• population dynamical• spatially implicit• unstructured• deterministic• based on differential equations
dF/dt = gmax F (1 – F/K) - N cref W2/3 F2/(F2 + H2)
dN/dt = - m N (1–W/Wmax)p/((1–W/Wmax)p + Lcritp)
dW/dt = e cref W2/3 F2 / (F2 + H2) – rref W2/3
The Scheffer et al. cohort-dynamics prey-predator model
• prey-predator• cohort-dynamics• spatially implicit• unstructured!!!• deterministic• based on differential equations
dF/dt = gmax F (1 – F/K) – N C
dN/dt = - m N Lp/(Lp + Lcritp)
dW/dt = e C - R
The Scheffer et al. cohort-dynamics prey-predator model (version 1.00)
• prey-predator• cohort-dynamics• spatially implicit• unstructured• deterministic• based on differential equations
dF/dt = gmax F (1 – F/K) – N C
Pmort(Ni) = - m Lp/(Lp + Lcritp)
dW/dt = e C - R
The Scheffer et al. cohort-dynamics prey-predator model (version 1.01)
• prey-predator• cohort-dynamics• spatially implicit• unstructured• stochastic• based on differential equations
dF/dt = gmax F (1 – F/K) – Ci
Pmort(Ni) = - m Lip/(Li
p + Lcritp)
dWi/dt = e Ci - Ri
The Scheffer et al. cohort-dynamics prey-predator model (version 1.02)
• prey-predator• cohort-dynamics• spatially implicit• individual-based (configuration with i individuals)• stochastic• based on differential equations
N
dF/dt = gmax F (1 – F/K) – Ci Ni
dNi/dt = - m Ni Lip/(Li
p + Lcritp)
dWi/dt = e Ci - Ri
The Scheffer et al. cohort-dynamics prey-predator model (version 1.03)
• prey-predator• cohort-dynamics• spatially implicit• individual-based (distribution with i classes)• deterministic• based on differential equations
N
The Scheffer et al. cohort-dynamics prey-predator model
deterministic stochastic
unstructured v1.00 v1.01
structured v1.03 v1.02
Advantages and disadvantages of each approach will be discussed during the exercises
Potential of each version
• v1.00 (unstr, determ): Periodic die offs
• v1.01 (unstr, stoch): Extinction probability
• v1.03 (dist, determ): Fitness of strains
• v1.02 (conf, stoch): Rate of strain replacement
Which model is best?
No ind. variation Ind. variation
No Moran effect Synchronized Not synchronized
Moran effect Synchronized Synchronized
Esa Ranta’s Large-scale synchronization model
Nine layers in model construction and analysis
• Language: C++, Visual-Basic
• Libraries: Runga-Kutta integration, etc.
• Frameworks: OSIRIS, EXCEL
• Type: Prey-predator cohort model
• Model: Scheffer et al. model
• Version: Unstr./config./distr., determ./stoch.
• Sensitivity: Parameter H
• Scenario: Brown trout
• Repetition: Demographic stochasticity
OSIRIS: Object-oriented Simulation Framework for Individual-based Simulations
“A personal solution to a shared problem”
Developed and used as a personal research tool, now also used in cooperation with others
Written in C++ (ca. 10000 lines)
Main aspects: predefined simulation objects, event queue, standardized input and output
Interface via EXCEL (through ASCII files)
Five predefined simulation objects
• Analyser (as many as there are analysis)
• System (one only)
• Habitat (as many as there are spatial units)
• Population (as many as there are species)
• Individual (as many as there are individuals)
Analyser
System
Habitat Population
Individual
datatasks
datatasks
datatasks
datatasks
datatasks
Simulating dynamics with an event queue
state at time t dynamics state at time t+t
task id 1 perform task 1 task id 3
task id 2 task id 2
task id 3 task id 1
task id 4 task id 4
task id 5 task id 5
OSIRIS library
PCLake code
Excel
Interface code
Borland C++
Input workbook
Notepad
Tools library
Log workbook
Output workbook
Report workbook
Input files
Output files
Executable