[R-sig-eco] solving differential equations for ecological models
Greetings - I'm having trouble figuring out an efficient means towards solving for the equilbria in a four-equation model with two herbivores (H1 and H2), and two predators (P and S). Below, I have set the rate of change for these populations to 0 (i.e., equilibrium), then solved. The resultant solutions are listed below. My goal now is to substitute all of these equations back into "H1", so as to get a general solution for the H1 equilibrium equation that is dependent only upon parametersobviously without doing it the long way. In sum, I'm looking to develop the equation "H1<-parameters" from the set of solutions below. Is this possible with deSolve in R? I'm relatively new to this package and despite much reading, can't figure out any efficient means as of yet. Advice appreciated, Peter H1<-1/alpha11-(alpha12*H2)/alpha11-(apred1*P)/(b1*alpha11)-(ashark1*S)/(b1*alpha11) H2<-1/alpha22-(alpha21*H1)/alpha22-(apred2*P)/(b2*alpha22)-(ashark2*S)/(b2*alpha22) S<-(apred1*epred1*H1)/ashark3+(apred2*epred2*H2)/ashark3-xpred/ashark3 P<-(eshark1*ashark1*H1)/(eshark3*ashark3)+(eshark2*ashark2*H2)/(eshark3*ashark3)-xshark/(eshark3*ashark3) -- Peter Houk, PhD Assistant Professor University of Guam Marine Laboratory http://www.guammarinelab.com/peterhouk.html www.pacmares.com www.micronesianfishing.com [[alternative HTML version deleted]] ___ R-sig-ecology mailing list R-sig-ecology@r-project.org https://stat.ethz.ch/mailman/listinfo/r-sig-ecology
Re: [R-sig-eco] solving differential equations for ecological models
Using Mathematica I get the following: H1 <- (apred1*ashark2*eshark2*(ashark3*b2+ashark2*xpred)+ ashark3*eshark3*(-(alpha22*b2*(ashark3*b1+ashark1*xpred))+ alpha12*b1*(ashark3*b2+ashark2*xpred))-alpha22*apred1*ashark3*b2*xshark+ apred2^2*ashark1*epred2*xshark-apred2*(-(ashark3* (ashark1*b2*epred2*eshark3+alpha12*b1*xshark))+ ashark2*(ashark3*b1*(eshark2+epred2*eshark3)+ ashark1*eshark2*xpred+apred1*epred2*xshark)))/ (apred2^2*ashark1^2*epred2*eshark1-apred1*apred2*ashark1*ashark2*epred2*eshark1- alpha11*apred2*ashark2*ashark3*b1*eshark2+alpha21*apred1*ashark2*ashark3*b2*eshark2- apred1*apred2*ashark1*ashark2*epred1*eshark2+apred1^2*ashark2^2*epred1*eshark2- alpha11*apred2*ashark2*ashark3*b1*epred2*eshark3+ alpha21*apred2*ashark1*ashark3*b2*epred2*eshark3+alpha12*ashark3*b1* (apred2*ashark1*eshark1+alpha21*ashark3*b2*eshark3+apred1*ashark2*epred1*eshark3)- alpha22*ashark3*b2*(alpha11*ashark3*b1*eshark3+apred1*ashark1*(eshark1+epred1*eshark3))) H2 <- (apred2*ashark1*eshark1*(ashark3*b1+ashark1*xpred)+ ashark3*eshark3*(alpha21*b2*(ashark3*b1+ashark1*xpred)- alpha11*b1*(ashark3*b2+ashark2*xpred))+apred1^2*ashark2*epred1*xshark- apred2*(alpha11*ashark3*b1+apred1*ashark1*epred1)*xshark- apred1*(ashark1*ashark3*b2*(eshark1+epred1*eshark3)+ ashark1*ashark2*eshark1*xpred-ashark3*(ashark2*b1*epred1*eshark3+alpha21*b2*xshark)))/ (apred2^2*ashark1^2*epred2*eshark1-apred1*apred2*ashark1*ashark2*epred2*eshark1- alpha11*apred2*ashark2*ashark3*b1*eshark2+ alpha21*apred1*ashark2*ashark3*b2*eshark2- apred1*apred2*ashark1*ashark2*epred1*eshark2+ apred1^2*ashark2^2*epred1*eshark2-alpha11*apred2*ashark2*ashark3*b1*epred2*eshark3+ alpha21*apred2*ashark1*ashark3*b2*epred2*eshark3+ alpha12*ashark3*b1*(apred2*ashark1*eshark1+alpha21*ashark3*b2*eshark3+ apred1*ashark2*epred1*eshark3)-alpha22*ashark3*b2* (alpha11*ashark3*b1*eshark3+apred1*ashark1*(eshark1+epred1*eshark3))) P <- (-(apred2*ashark1*ashark2*b1*epred2*eshark1)+ apred2*ashark1^2*b2*epred2*eshark1- alpha11*ashark2*ashark3*b1*b2*eshark2+ alpha21*ashark2*ashark3*b1*b2*eshark2+ apred1*ashark2^2*b1*epred1*eshark2- apred1*ashark1*ashark2*b2*epred1*eshark2- alpha11*ashark2^2*b1*eshark2*xpred+ alpha21*ashark1*ashark2*b2*eshark2*xpred+ apred2*(alpha11*ashark2*b1-alpha21*ashark1*b2)*epred2* xshark+alpha22*b2* (-(ashark1*eshark1*(ashark3*b1+ashark1*xpred))+ (alpha11*ashark3*b1+apred1*ashark1*epred1)*xshark)+ alpha12*b1*(ashark1*eshark1*(ashark3*b2+ashark2*xpred)- (alpha21*ashark3*b2+apred1*ashark2*epred1)*xshark))/ (apred2^2*ashark1^2*epred2*eshark1- apred1*apred2*ashark1*ashark2*epred2*eshark1- alpha11*apred2*ashark2*ashark3*b1*eshark2+ alpha21*apred1*ashark2*ashark3*b2*eshark2- apred1*apred2*ashark1*ashark2*epred1*eshark2+ apred1^2*ashark2^2*epred1*eshark2- alpha11*apred2*ashark2*ashark3*b1*epred2*eshark3+ alpha21*apred2*ashark1*ashark3*b2*epred2*eshark3+ alpha12*ashark3*b1* (apred2*ashark1*eshark1+alpha21*ashark3*b2*eshark3+ apred1*ashark2*epred1*eshark3)- alpha22*ashark3*b2* (alpha11*ashark3*b1*eshark3+ apred1*ashark1*(eshark1+epred1*eshark3))) S <- (apred2^2*b1*epred2*(ashark1*eshark1-alpha11*xshark)+ b2*((alpha12-alpha22)*apred1*ashark3*b1*epred1*eshark3+ apred1*(alpha22*ashark1*eshark1- alpha21*ashark2*eshark2)*xpred+ (-(alpha12*alpha21)+alpha11*alpha22)*ashark3*b1*eshark3* xpred+apred1^2*epred1* (ashark2*eshark2-alpha22*xshark))+ apred2*(b1*((-alpha11+alpha21)*ashark3*b2*epred2*eshark3- alpha12*ashark1*eshark1*xpred+ alpha11*ashark2*eshark2*xpred)+ apred1*(-(ashark1*b2*epred2*eshark1)- ashark2*b1*epred1*eshark2+alpha12*b1*epred1*xshark+ alpha21*b2*epred2*xshark)))/ (apred2^2*ashark1^2*epred2*eshark1- apred1*apred2*ashark1*ashark2*epred2*eshark1- alpha11*apred2*ashark2*ashark3*b1*eshark2+ alpha21*apred1*ashark2*ashark3*b2*eshark2- apred1*apred2*ashark1*ashark2*epred1*eshark2+ apred1^2*ashark2^2*epred1*eshark2- alpha11*apred2*ashark2*ashark3*b1*epred2*eshark3+ alpha21*apred2*ashark1*ashark3*b2*epred2*eshark3+ alpha12*ashark3*b1* (apred2*ashark1*eshark1+alpha21*ashark3*b2*eshark3+ apred1*ashark2*epred1*eshark3)- alpha22*ashark3*b2* (alpha11*ashark3*b1*eshark3+ apred1*ashark1*(eshark1+epred1*eshark3))) Jim Jim Baldwin Station Statistician Pacific Southwest Research Station USDA Forest Service -Original Message- From: r-sig-ecology-boun...@r-project.org [mailto:r-sig-ecology-boun...@r-project.org] On Behalf Of peterhouk1 . Sent: Wednesday, December 25, 2013 8:57 PM To: r-sig-ecology@r-project.org Subject: [R-sig-eco] solving differential equations for ecological models Greetings - I'm having trouble figuring out an efficient means towards solving for the equilbria in a four-equation model with two herbivores (H1 and H2), and two predators (P and S). Below, I have set the rate of change for these populations to 0 (i.e., equilibrium), then solved. The resultant solutions are listed below. My goal now is to substitute all of these equations back into "H1", so as to get a general solution for the H1 equilibrium equation that is dependent only upon paramete
