#Further Excercise 5
var("P")
var("N")
Nprime=0.5*N-0.1*N*P
Pprime=0.5*0.01*N*P-0.2*P
NP_vector = (Nprime,Pprime)
plot_vector_field(NP_vector, (N,0,10), (P,0,10), axes_labels=["N","P"])

t=srange(0,100,0.1)
sol1=desolve_odeint([Nprime,Pprime],ics=[1,3],dvars=[N,P],times=t)
list_plot(zip(t,sol1[:,0]), plotjoined=true)+list_plot(zip(t,sol1[:,1]), color="red", plotjoined=true)

t=srange(0,100,0.1)
sol2=desolve_odeint([Nprime,Pprime],ics=[0.2,1],dvars=[N,P],times=t)
list_plot(zip(t,sol2[:,0]), plotjoined=true)+list_plot(zip(t,sol2[:,1]), color="red", plotjoined=true)

plot_vector_field(NP_vector, (N,0,400), (P,0,30), axes_labels=["N","P"])+list_plot(sol1,plotjoined=true, color="blue")+list_plot(sol2, plotjoined=true, color="magenta")

#b
#The Holling-Tanner model is a much more defined version of the Lokta Volterra model. The LV model does not set realistic expectations of the behavior of a model while the Holling-tanner model incoporates limitations that help make the system more biologically sound- like adding a carrying capacity to the naturalk growth of populations or adding a saturation point to the amount of prey the predator can eat.






var("G","H","P","n") #set variable 
k1=0.2 # define values
k2=0.21
k3=0.18
t=srange(0,100,0.1) #set srange
Hprime=(1/(1+G^n))-k1*H
@interact
def nval(n=(2,12)):
     Hprime=(1/(1+G^n))-k1*H #define differential equations 
P2prime=H-k2*P
Gprime=P-k3*G
system=[Hprime,Gprime, P2prime] #make all differential equations in one list
sol5=desolve_odeint(system,ics=[3,5,4],dvars=[H,G,P],times=t) #use desolve
p=list_plot(zip(t,sol5[:,0]), plotjoined=true, color="lime")+list_plot(zip(t,sol5[:,1]), color="magenta", plotjoined=true)+list_plot(zip(t,sol5[:,2]), color="blue",plotjoined=true)
show(p)