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Question: Check positivity

Question:Check positivity

Dkunb 65 Maple 2021
assumptions assuming
March 17 2023
0

Why did the last line is(ysol2[1]>0) give false?

Maple does not recognize the assumption?

Thanks in advance.
 

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restart;
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interface(showassumed=0);

0

 (1)
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assume(theta>1,alpha>0,sigma1>0,beta>0,sigma2>0,delta>0,x>0,y>0,z>0);
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f:=((theta*x-1)*(1-x)-y);
g:=y/(1+beta*y^2);
h:=(alpha*x-sigma1)*(1+beta*y^2);
j:=(delta*g-sigma2);
dxdt:=x*f;
dydt:=g*(h-z);
dzdt:=z*j;

(theta*x-1)*(1-x)-y

  

y/(beta*y^2+1)

  

(alpha*x-sigma1)*(beta*y^2+1)

  

delta*y/(beta*y^2+1)-sigma2

  

x*((theta*x-1)*(1-x)-y)

  

y*((alpha*x-sigma1)*(beta*y^2+1)-z)/(beta*y^2+1)

  

z*(delta*y/(beta*y^2+1)-sigma2)

 (2)

case0; x = 0, y = 0, z = 0

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E0:=<0,0,0>;

Vector[column](%id = 36893490685786064044)

 (3)

"case1: z=0 , h(x,y)=0, f(x,y)=0 : From y=(theta*x-1)*(1-x)=y -> 1/(theta)<x and x<1"

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xsol1:=solve(h=0,x) assuming x>1/theta and x<1;
ysol1:=solve(subs(x=xsol1,f)=0,y)assuming xsol1>1/theta and xsol1<1;
E1:=<xsol1,ysol1,0>;

xsol1 := sigma1/alpha

  

ysol1 := -(-sigma1*theta+alpha)*(alpha-sigma1)/alpha^2

  

Vector[column](%id = 36893490685786055012)

 (4)

case2; g(y) = 0, h(x, y) = 0

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ysol:=[solve(j=0,y)];
ysol2:=simplify(subs(sigma2=delta*eta,ysol)) assuming (1-4*beta*eta^2>0 and eta>0);

Warning, solve may be ignoring assumptions on the input variables.

  

[(1/2)*(delta+(-4*beta*sigma2^2+delta^2)^(1/2))/(beta*sigma2), -(1/2)*(-delta+(-4*beta*sigma2^2+delta^2)^(1/2))/(beta*sigma2)]

  

[(1/2)*(1+(-4*beta*eta^2+1)^(1/2))/(beta*eta), (1/2)*(1-(-4*beta*eta^2+1)^(1/2))/(beta*eta)]

 (5)
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is(ysol2[1]>0)

false

 (6)
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(1/2)*(1+(-4*beta*eta^2+1)^(1/2))/(beta*eta)

 (7)
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