Harvested prey–predator model incorporating a prey refuge

Question

I want to plot the phase diagram of prey predator versus prey refuge to see how the prey refuge influences the population of prey and predator. And this is the system

$x'=\alpha x(1-x/k)-\beta\frac{(1-m)xy}{1+a(1-m)x}-q_1E_1x$

$y'=-\gamma y+c\beta\frac{(1-m)xy}{1+a(1-m)x}-q_2E_2y$

The prey predator with Holling type II model is incorporating a prey refuge, $mx$ and $k$, $\alpha$, $\gamma$, $c$ and $\beta/\alpha$ are the carrying capacity, growth rate of prey, death rate of predator, conversion factor denoting the number of newly born predators for each captured prey and maximum number of prey that can be eaten by each predator in unit time respectively.

I have the numerical value for $a=0.02$, $k=100$, $\alpha=10$, $\beta=0.6$, $\gamma=0.09$, $c=0.02$.

Thanks so much!

Answer 1

I solved more general system linked by @SjoerdC.deVries in the comments reproducing figure 3 and 4 - to prove it is correct. You can simplify this to version you need.

Clear["Global`*"]

al = 2;
a = 2/1000;
k = 600;
b = 1/10;
g = 46/10^5;
c = 1/100;
m = 1/100;
E1 = 1;
q1 = 2/10;
E2 = 813/1000;
q2 = 2/100;
Tf = 300;

eqs = {
   x'[t] == 
    al x[t] (1 - x[t]/k) - b (1 - m) x[t] y[t]/(1 + a (1 - m) x[t]) - 
     q1 E1 x[t],
   y'[t] == -g y[t] + c b (1 - m) x[t] y[t]/(1 + a (1 - m) x[t]) - 
     q2 E2 y[t],
   x[0] == 2, y[0] == 8};

s = NDSolve[eqs, {x, y}, {t, Tf}];

Plot[Evaluate[{x[t], y[t]} /. s], {t, 0, Tf}, PlotStyle -> Automatic, 
 ImageSize -> 300, PlotRange -> All, Frame -> True]
ParametricPlot[Evaluate[{x[t], y[t]} /. s], {t, 0, Tf}, 
 AspectRatio -> 1, PlotRange -> All, ImageSize -> 300, Frame -> True]

Answer 2

Some modifications for the code in Vitaliy Kaurov's answer:

<<MaTeX`
<< c:\CurvesGraphics6\CurvesGraphics6.m

Initial condition outside the limit cycle:

α = 200/207; 
a = 4; 
k = 3; 
β = 200/69 + 1/50; 
m = 1/2; 
c = 4761/20000; 
γ = 1/10; 
E1 = 500/621; 
q1 = 2/10; 
E2 = 3/4; 
q2 = 2/100; 
Tf = 1000; 
eqs1 = {x'[t] == -E1 q1 x[t] + x[t] (1 - x[t]/k) α - ((1 - m) x[t]*y[t] β)/(1 + a (1 - m) x[t]), 
y'[t] == -E2 q2 y[t] + (c (1 - m) x[t]*y[t] β)/(1 + a (1 - m) x[t]) - y[t]γ, x[0] == 1.1, y[0] == 1.1};
s1 = NDSolve[eqs1, {x, y}, {t, Tf}];

ts1 = Plot[Evaluate[{x[t], y[t]} /. First[s1]], {t, 0, Tf}, 
PlotRange -> All, 
PlotStyle -> {{Blue, Thickness[0.003]}, {Red, Thickness[0.003]}}, 
Frame -> True, 
FrameLabel -> { MaTeX["\\text{t}", Magnification -> 1], 
MaTeX["\\text{Population densities}", Magnification -> 1]}, 
LabelStyle -> Directive[Black, Tiny], RotateLabel -> True, 
PlotPoints -> 500, ImageSize -> {300, 200}, 
DisplayFunction -> Identity]

pp1 = ParametricPlot[Evaluate[{x[t], y[t]} /. s1], {t, 0, 100}, 
Axes -> False, Oriented -> True, ArrowPositions -> {0.05, 0.985}, 
PlotStyle -> {Black, Thickness[0.002], Arrowheads[0.035]}, 
AspectRatio -> 1, PlotRange -> All, Frame -> True, 
RotateLabel -> True, 
FrameLabel -> { 
MaTeX["\\text{Prey population density}", Magnification -> 1], 
MaTeX["\\text{Predator population density}", Magnification -> 1]},
RotateLabel -> False, LabelStyle -> Directive[Black, Tiny], 
ImageSize -> {250, 250}]

Initial condition inside the limit cycle:

α = 200/207; 
a = 4; 
k = 3; 
β = 200/69 + 1/50; 
m = 1/2; 
c = 4761/20000; 
γ = 1/10; 
E1 = 500/621; 
q1 = 2/10; 
E2 = 3/4; 
q2 = 2/100; 
Tf = 1000;
eqs2 = {x'[t] == -E1 q1 x[t] + x[t] (1 - x[t]/k) α - ((1 - m) x[t]*y[t] β)/(1 + a (1 - m) x[t]), 
y'[t] == -E2 q2 y[t] + (c (1 - m) x[t]*y[t] β)/(1 + a (1 - m) x[t]) - y[t]γ, x[0] == 1.02, y[0] == 1.02};
s2 = NDSolve[eqs2, {x, y}, {t, Tf}];

ts2 = Plot[Evaluate[{x[t], y[t]} /. First[s2]], {t, 0, Tf}, 
PlotRange -> All, 
PlotStyle -> {{Blue, Thickness[0.003]}, {Red, Thickness[0.003]}}, 
Frame -> True, 
FrameLabel -> { MaTeX["\\text{t}", Magnification -> 1], 
MaTeX["\\text{Population densities}", Magnification -> 1]}, 
LabelStyle -> Directive[Black, Tiny], RotateLabel -> True, 
PlotPoints -> 500, ImageSize -> {300, 300}, 
DisplayFunction -> Identity]

pp2 = ParametricPlot[Evaluate[{x[t], y[t]} /. s2], {t, 900, Tf}, 
PlotStyle -> {Black, Thickness[0.008]}, AspectRatio -> 1, 
PlotRange -> All, Frame -> True, RotateLabel -> True, 
FrameLabel -> { 
MaTeX["\\text{Prey population density}", Magnification -> 1], 
MaTeX["\\text{Predator population density}", Magnification -> 1]},
RotateLabel -> False, LabelStyle -> Directive[Black, Tiny], 
ImageSize -> {250, 250}] 

The initial conditions and equilibrium point:

p0 = Part[{x, y} /. NSolve[{-E1 q1 x + x (1 - x/k) α - ((1 - m) x*y β)/(1 + a (1 - m) x), -E2 q2 y + (c (1 - m) x*y β)/(1 + a (1 - m) x) - y γ} == {0, 0} && x > 0 && y > 0, {x, y}], 1];
(*{0.97972, 0.992966}*)
IC1 = {1.02, 1.02};
IC2 = {1.1, 1.1};

pointIC1 = 
Graphics[{PointSize[0.015], Darker[Green], Point[IC1]}, 
IC1 - {-0.08, -0.011}, Axes -> True, PlotRange -> All, 
ImageSize -> {250, 250}];

pointIC2 = 
Graphics[{PointSize[0.015], Darker[Green], Point[IC2]}, 
IC2 - {-0.08, -0.011}, Axes -> True, PlotRange -> All, 
ImageSize -> {250, 250}];

Finally, when applying the command Show produces

Show[pp2, pp21, pp1, pointIC1, pointIC2, 
Epilog -> {{PointSize[0.018], Black, Point[p0]}, {PointSize[0.013], 
White, Point[p0]}}, Axes -> False, ImageSize -> {350, 350}]