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El programa$$\PageIndex{1}$$ simula la cosecha con el llamado rendimiento máximo sustentable. Introduce pequeñas fluctuaciones aleatorias en la población, tan pequeñas que no se pueden discernir en una gráfica. La ligera estocástica hace que el programa tome una trayectoria diferente cada vez que se ejecuta, con cursos de tiempo muy diferentes. Inevitablemente, sin embargo, las poblaciones se desploman por debajo del punto de Allee y colapsan rápidamente, como en la ejecución muestral del programa mostrado en la Figura$$\PageIndex{1}$$.

En la época de la navegación, en la flecha marcada con “A”, la pesca era de alto esfuerzo pero de bajo impacto y las pesquerías se mantuvieron aproximadamente a su capacidad de carga,$$K$$. La “cosecha óptima” se introdujo una vez que la ecología matemática combinada con la tecnología diesel, y la pesca ayudó a alimentar a las crecientes poblaciones humanas y animales domésticos, con poblaciones de peces cerca del “rendimiento máximo sustentable”, como se esperaba. Pero a lo largo del siglo XX, como se muestra a ambos lados de la flecha marcada con “B”, las poblaciones de peces continuaron disminuyendo, y antes de 2015 —en la flecha marcada con “C ”— queda claro que algo anda muy mal.

# SIMULATE ONE YEAR
#
# This routine simulates a differential equation for optimal harvesting
# through one time unit, such as one year, taking very small time steps
# along the way.
#
# The ’runif’ function applies random noise to the population. Therefore it
# runs differently each time and the collapse can be rapid or delayed.
#
# ENTRY: ’N’ is the starting population for the species being simulated.
#        ’H’ is the harvesting intensity, 0 to 1.
#        ’K’ is the carrying capacity of the species in question.
#        ’r’ is the intrinsic growth rate.
#        ’dt’ is the duration of each small time step to be taken throughout
#           the year or other time unit.
#
# EXIT:  ’N’ is the estimated population at the end of the time unit.

SimulateOneYear = function(dt)
{ for(v in 1:(1/dt))                  # Advance the time step.
{ dN = (r+s*N)*N - H*r^2/(4*s)*dt;  # Compute the change.
N=N+dN; }                         # Update the population value.
if(N<=0) stop("Extinction");        # Make sure it is not extinct.
assign("N",N, envir=.GlobalEnv); }  # Export the results.

r=1.75; s=-0.00175; N=1000; H=0;      # Establish parameters.

for(t in 1850:2100)                   # Advance to the next year.
{ if(t>=1900) H=1;                    # Harvesting lightly until 1990.
print(c(t,N));                      # Display intermediate results.
N = (runif(1)*2-1)*10 + N;          # Apply stochasticity.
SimulateOneYear(1/(365*24)); }      # Advance the year and repeat.


Programa$$\PageIndex{1}$$. Este programa simula la cosecha máxima con pequeñas fluctuaciones en las poblaciones.

¿Qué pasó? Un colapso es parte de la dinámica de este tipo de cosechas. La estocástica inevitable en la cosecha se combina desfavorablemente con un equilibrio inestable en la población de presas. En algunas carreras se derrumba en 80 años, en otras puede tardar 300. El tiempo no es predecible; la principal propiedad predecible de la simulación es que en última instancia el sistema colapsará.

This page titled 13.3: Modelado estocástico is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Clarence Lehman, Shelby Loberg, & Adam Clark (University of Minnesota Libraries Publishing) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.