Deterministické modely růstu a existenčního boje

Modely růstu

> unassign('x','t','a','g','k');

Nechť  značí velikost růstu populace v čase . Předpokládáe, že  plňuje rovnici:

> r[1]:=diff(x(t),t)=a*x(t)*g(x(t)/k);

pro nějkou funkci  a konstanty  a .

Rovnice  vyjadřuje, jak rychlost růstu populce závisí na její velikosti

Malthusův předpoklad

Za předpokladu, že funkce je konstantní

g:=y->1;


má rovnice tvar:

Malt:=r[1];


a její řešení je:

M:=dsolve(Malt);

g:='g':

> C1:=solve(subs(t=0,a=.4,k=1,rhs(M))=10);

> plot(subs(_C1=C1,a=.4,rhs(M)),t=0..10);

nakreslíme řešení pro různé počáteční podmínky:

> with(DEtools);

ivs:=[seq(x(0)=i*10,i=-1..5)];

DEplot( subs(a=.4,Malt), x(t), t=0..10, ivs );

#phaseportrait( subs(a=.4,Malt), x(t), t=0..10, ivs );


Verhulstův přdpoklad  (logistická křivka)

> g:=y->1-y;

Verh:=r[1];

V:=dsolve(Verh);

g:='g':

H. Hotelling, stanovil v roce 1940 hodnotu parametr; a=0.031239, b=67.5352, k=  pro po4et obzvatel
v USA.

> C1:=solve(subs(t=0,a=.4,k=1,rhs(V))=1);

> plot(subs(_C1=10,a=.4,k=1,rhs(V)),t=0..10);

> InlexniBod=solve(

simplify(diff(rhs(V),t,t))=0

,t);

> assume(k>0,a>0,_C1>0);

#with(RealDomain);

Limit(rhs(V),t=infinity)=limit(rhs(V),t=infinity);

> with(plots);

p1:=plot(subs(_C1=10,a=.4,k=1,{rhs(V),k}),t=0..20):

p2:=textplot([17,1.02,"saturační hranice"],align={above,right}):

display({p1,p2});

unassign('k','a','_C1');


> ivs:=[seq(x(0)=i*.1,i=0..15)];

DEplot( subs(a=.4,k=1,Verh), x(t), t=0..10, ivs, animatecurves = true );

>

Gompertzův předpoklad

> g:=y->-ln(y);

with(PDEtools,declare);

declare(x(t),prime=t);;

Gomp:=r[1];

infolevel[dsolve] := 3;

#odeadvisor(Gomp, help);

G:=simplify(dsolve(Gomp));

g:='g':

Methods for first order ODEs:

--- Trying classification methods ---

trying a quadrature

trying 1st order linear

trying Bernoulli

trying separable

<- separable successful

> C1:=solve(subs(t=0,a=.4,k=1,rhs(G))=.2);

> with(RealDomain);

G2:=dsolve({Gomp,x(0)=.1});

Methods for first order ODEs:

--- Trying classification methods ---

trying a quadrature

trying 1st order linear

trying Bernoulli

trying separable

<- separable successful

> plot(subs(_C1=10,a=.4,k=1,_Z2=10,{rhs(G),rhs(G)}),t=0..10);

> infolevel[dsolve] := 0;

ivs:=[seq(x(0)=i*.1,i=0..15)];

DEplot( subs(a=.4,k=1,Gomp), x(t), t=0..10, ivs, animatecurves = true );

Warning, plot may be incomplete, the following errors(s) were issued:

   cannot evaluate the solution past the initial point, problem may be complex, initially singular
or improperly set up

> xxx:=subs(a=.4,k=1,Gomp);

zzz:=dsolve({xxx,x(0)=.1},numeric);

> f:=t->op(2,zzz(t));

> plot(rhs(zzz(t)[2]),t=0..10);

Error, invalid input: rhs received zzz(t)[2], which is not valid for its 1st argument, expr


> aaa:=rhs(zzz(t)[2]);

>

Error, invalid input: rhs received zzz(t)[2], which is not valid for its 1st argument, expr


> plot(aaa,t=0..10);

Warning, unable to evaluate the function to numeric values in the region; see the plotting
command's help page to ensure the calling sequence is correct


Plotting error, empty plot

>

Dvě populace soupeřící o obživu

> with(DEtools);unassign('x','t','a','k','c','_r');

> _r[1]:=diff(x[1](t),t)=a[1]*x[1](t)*(k[1]-x[1](t)-c[1]*x[2](t))/k[1];

_r[2]:=diff(x[2](t),t)=a[2]*x[2](t)*(k[2]-x[2](t)-c[2]*x[1](t))/k[2];

> dfieldplot(subs(k[1]=105,k[2]=64,a[1]=1.124,a[2]=.794,c[1]=1.64,c[2]=.61,[_r[1],_r[2]]),

[x[1](t),x[2](t)],t=-2..2, x[1]=39..40.5, x[2]=39..40.5, arrows=SMALL,

title=``, color=[.3*x[2](t)*(x[1](t)-1),x[1](t)*(1-x[2](t)),.1]);

> A:='A';B:='B';

xxx:=solve({k[2]=A,k[1]/c[1]=B,k[2]/c[2]=E,k[1]=F},{k[1],k[2],c[1],c[2]});

>

> dfieldplot(subs(

subs(A=1,B=2,E=1,F=2,xxx),

a[1]=2.,a[2]=1,

[_r[1],_r[2]]),

[x[1](t),x[2](t)],t=-2..2, x[1]=-.2..3, x[2]=-.1..2, arrows=SMALL,

title=`Limitni stav je [k[1],0]`, color=[.3*x[2](t)*(x[1](t)-1),x[1](t)*(1-x[2](t)),.1]);


dfieldplot(subs(

subs(A=2,B=1,E=1,F=2,xxx),

a[1]=2.,a[2]=1,

[_r[1],_r[2]]),

[x[1](t),x[2](t)],t=-2..2, x[1]=-.2..2.3, x[2]=-.1..2.3, arrows=SMALL,

title=`Limitni stavy [k[1],0], nebo [0,k[2]] zaviseji na poc. podm.`,
color=[.3*x[2](t)*(x[1](t)-1),x[1](t)*(1-x[2](t)),.1]);


dfieldplot(subs(

subs(A=1,B=2,E=2,F=1,xxx),

a[1]=2.,a[2]=1,

[_r[1],_r[2]]),

[x[1](t),x[2](t)],t=-2..2, x[1]=.4..0.8, x[2]=0.5..1.2, arrows=SMALL,

title=`Limitni stav je [(k[1]-c[1]*k[2])/(1-c[1]*c[2]),(k[2]-c[2]*k[1])/(1-c[1]*c[2])]`,
color=[.3*x[2](t)*(x[1](t)-1),x[1](t)*(1-x[2](t)),.1]);



dfieldplot(subs(

subs(A=2,B=1,E=2,F=1,xxx),

a[1]=2.,a[2]=1,

[_r[1],_r[2]]),

[x[1](t),x[2](t)],t=-.2..2, x[1]=-.2..2.1, x[2]=-.1..2, arrows=SMALL,

title=`Limitni stav je [0,k[2]]`, color=[.3*x[2](t)*(x[1](t)-1),x[1](t)*(1-x[2](t)),.1]);



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