##### Model Description

This is a simple model of a prey–predator system in two areas, one of which receives fishing activity (fishing ground) and another that does not (MPA: marine protected area). Takashina et al. 2012 gives the following description of the model. "A number of fish and invertebrate stocks have been depleted by overexploitation in recent years. To address this, marine protected areas (MPAs) are often established to protect biodiversity and recover stocks. We analyzed the potential impact of establishing MPAs on marine ecosystems using mathematical models. We demonstrate that establishment of an MPA can sometimes result in a considerable decline, or even extinction, of a species. We focus on a prey–predator system in two patches, one exposed to fishing activity and the other protected (MPA). Our analyses reveal that the establishment of the MPA can cause a reduction in prey abundance, and even extinction of the prey. Such unintended consequences are more likely to occur if the predator species is a generalist and if the MPA is intended to protect only the predatory species".

**Reference**

Takashina, N., Mougi, A., & Iwasa, Y. (2012). Paradox of marine protected areas: suppression of fishing may cause species loss. Population Ecology, 54(3), 475-485. doi:10.1007/s10144-012-0323-8

##### Scenarios

Two cases are explored here. In the first case, (1) both prey and predator are harvested in the open fishing area, (2) both are protected in the MPA, and (3) predator species is generalist (i.e., r_{y}>0). We also assume that the condition of 'trophic cascade effect' is satisfied here (see the inequality expression in the equations section below). The paradox is that increasing MPA (R) can unintendedly cause the extinction of prey species. To observe this, set r_{x}=2, r_{y}=0.9, K_{x}=40, K_{y}=10, q_{x}=1, q_{y}=1, E_{x}=0.7, E_{y}=0.7, a=0.3, theta=0.3, c_{x}=0.1, c_{y}=0.1, m_{x}=0.1, and m_{y}=1. Also set the initial populations of prey and predator in the open area (x_{1} and y_{1} respectively) and in the MPA (x_{2} and y_{2} respectively) to some value (e.g., 0.2). Now vary the protected area (R) from 0 to 1 and track how the combined prey population (x_{1}+x_{2}) changes. Set the y-axis to the auxilary variable 'prey' and the x-axis to time to see this dynamics. You will see that at around R=0.5, the prey population becomes extinct.

In the second case, (1) both prey and predator are harvested in the open fishing area, (2) both of them are protected in the MPA, and (3) predator species is specialist (i.e., r_{y}=0). The interesting observation here is that when predators are specialists, prey species will never go extinct. To see this, set r_{x}=2, r_{y}=0, K_{x}=40, K_{y}=10, q_{x}=1, q_{y}=1, E_{x}=0.7, E_{y}=0.7, a=0.3, theta=0.3, c_{x}=0.1, c_{y}=0.1, m_{x}=0.1, and m_{y}=1. Now vary the protected area (R) from 0 to 1 and track how the combined prey population (x_{1}+x_{2}) changes. You will see that, although approaching extinction, prey will never actually reach extinction.

$\Large \frac{dx_{1}}{dt}=r_{x}x_{1}(1-\frac{x_{1}}{(1-R)K_{x}})-\frac{ax_{1}y_{1}}{1-R}-c_{x}x_{1}-q_{x}E_{x}x_{1}+m_{x}\left [(1-R)x_{2}-Rx_{1} \right ]$ |

Rate of prey population change in area 1 (open fishing area) |

$\Large \frac{dx_{2}}{dt}=r_{x}x_{2}(1-\frac{x_{2}}{RK_{x}})-\frac{ax_{2}y_{2}}{R}-c_{x}x_{2}+m_{x}\left [Rx_{1}-(1-R)x_{2} \right ]$ |

Rate of prey population change in area 2 (marine protected area) |

$\Large \frac{dy_{1}}{dt}=r_{y}y_{1}(1-\frac{y_{1}}{(1-R)K_{y}})-\frac{\theta ax_{1}y_{1}}{1-R}-c_{y}y_{1}-q_{y}E_{y}y_{1}+m_{y}\left [(1-R)y_{2}-Ry_{1} \right ]$ |

Rate of predator population change in area 1 (open fishing area) |

$\Large \frac{dy_{2}}{dt}=r_{y}y_{2}(1-\frac{y_{2}}{RK_{y}})-\frac{\theta ax_{2}y_{2}}{R}-c_{y}y_{2}+m_{y}\left [Ry_{1}-(1-R)y_{2} \right ]$ |

Rate of predator population change in area 2 (marine protected area) |

$\Large trophic \ cascade=\frac{aK_{y}}{r_{y}}-\frac{q_{x}E_{x}}{q_{y}E_{y}}>0$ |

If this expression holds, trophic cascade applies (the paradox of prey population reduction upon the introduction of marine protected area can happen). |

#=====define parameters par rx=2, ry=0.9, Kx=40, Ky=10, qx=1, qy=1, Ex=0.7, Ey=0.7, a=0.3, theta=0.3, cx=0.1, cy=0.1 par mx=1, my=2, R=0.1 #=====define some hidden variables===== #============economic functions #==================demographicfunctions #=====auxiliary quantities================= aux tropcas=a*Ky/ry-(qx*Ex)/(qy*Ey) aux ratio=my/mx aux prey=x1+x2 aux pred=y1+y2 #======right hand sides dx1/dt=rx*x1*(1-x1/((1-R)*Kx))-a*x1*y1/(1-R)-cx*x1-qx*Ex*x1+mx*((1-R)*x2-R*x1) dx2/dt=rx*x2*(1-x2/(R*Kx))-a*x2*y2/(R)-cx*x2+mx*(R*x1-(1-R)*x2) dy1/dt=ry*y1*(1-y1/((1-R)*Ky))+theta*a*x1*y1/(1-R)-cy*y1-qy*Ey*y1+my*((1-R)*y2-R*y1) dy2/dt=ry*y2*(1-y2/(R*ky))+theta*a*x2*y2/(R)-cy*y2+my*(R*y1-(1-R)*y2) #=============initial data init x1=0.2, x2=0.2, y1=0.2, y2=0.2 @ meth=qualrk @ bound=1000 @ total=100 @ dt=0.5 @ yp=x1 @ ylo=0,yhi=30 done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.