Critical points of a system

I have the following system:

For this system I have to calculate the three equilibria (critical points). Here are the equations in Mathematica:

eqH = HH'[t] == (bH/NN)*(NN - CC[t] - HH[t])*HH[t] - (sH*aH)*HH[t]
eqC = CC'[t] == (bC/NN)*(NN - CC[t] - HH[t])*CC[t] - (sC + aC)*CC[t]
eqS = SS[t] == NN - HH[t] - CC[t]


L have the following values for the constants:

NN := 400
sS := 1/5
sC := 1/7
sH := 1/5
bC := 0.45
bH := 0.4
aC := 0.1
aH := 0.1


But now I don't know what to do next. For critical points I don't know how to solve the equations H'[t] == 0, C'[t] == 0, S[t] == 0

• Have you seen Solve? Feb 4 '18 at 23:18
• @MichaelE2 I tried to solve the system like this: system = {eqH, eqC} , solSystem = DSolve[system, {hh, cc}, t] but it's not ok Feb 5 '18 at 0:05
• You don't want to set S[t]==0 but you do need to set eqH and eqC equal to zero in Solve. Don't use DSolve since you're solving algebraic equations. Feb 5 '18 at 2:40

Block[{Nc, Sc, dC, dH, bC, bH, aC, aH, St, Ht, Ct},

With[{$Nc = 400,$Sc = 0.2, $dC = 1./7,$dH = 0.2, $bC = 0.45,$bH = 0.4, $aC = 0.1,$aH = 0.1},
With[{paramsRl = Thread[{Nc, Sc, dC, dH, bC, bH, aC, aH} -> {$Nc,$Sc, $dC,$dH, $bC,$bH, $aC,$aH}]},

With[{xRHS = (bH/Nc) (Nc - Ct - Ht) Ht - (dH + aH) Ht},
With[{yRHS = (bC/Nc) (Nc - Ct - Ht) Ct - (dC + aC) Ct},
With[{CST = Sc - Nc + Ht + Ct},

Block[{horIsoclRl, vertIsoclRl, critPtsRl, expr},
(* horizontal isoclines ie dC/dH \[Equal] 0 *)
horIsoclRl = Solve[yRHS == 0, Ct];

(* vertical isoclines ie dC/dH \[Equal] Infintiy *)
vertIsoclRl = Solve[xRHS == 0, Ct];

(* critical points ie Ht' \[Equal] 0 && Ct' \[Equal] 0 *)
critPtsRl = Solve[{xRHS == 0, yRHS == 0}, {Ht, Ct}];

Block[{isoclGraph, critPtsGraph, cstGraph, critPtsTab, paramsTab, C, H},

(* plot isoclines - horizontal: Gray, vertical: Blue *)
With[{xMn = -150, xMx = 350},
isoclGraph = MapIndexed[
Graphics[
If[
#2[[1]] == 1,
{Gray, Thin, Opacity[0.5],
Line[{{xMn, #1[[-1]] /. Ht -> xMn}, {xMx, #1[[-1]] /. Ht -> xMx}}]},
{Lighter[Blue], Thin, Opacity[0.5], Line[{{xMn, #1[[-1]] /. Ht -> xMn}, {xMx, #1[[-1]] /. Ht -> xMx}}]}
] /. paramsRl] &,
{{Ht, Ct} /. horIsoclRl, {Ht, Ct} /. vertIsoclRl},
{2}
];

(* plot critical points - Red *)
critPtsGraph = Graphics[
{PointSize[Large], Darker[Red], Point[#]}
] & /@ (({Ht, Ct} /. critPtsRl) /. paramsRl);

(* plot constraint - Brown *)
cstGraph = MapIndexed[
Graphics[
{Brown, Thin, Dashed, Opacity[0.7],
Line[{{xMn, #1[[-1]] /. Ht -> xMn}, {xMx, #1[[-1]] /. Ht -> xMx}}]} /. paramsRl] &, (({Ht, Ct} /. Solve[CST == 0, Ct]) /. paramsRl)];

(* create critical points' table *)
critPtsTab = Grid[

Partition[
Style[{Ht, Ct}, Darker[Blue]] /. critPtsRl /. paramsRl, 2,
2, {1, 1}, {}],
Alignment -> Left
];

(* create parameters' table *)
paramsTab = Grid[
Prepend[
Partition[

Style[#, Gray, 12] & /@ {N, S, Subscript[δ, C],
Subscript[δ, H], Subscript[β, C], Subscript[β, H],
Subscript[α, C], Subscript[α, H]} -> {$Nc,$Sc, $dC,$dH, $bC,$bH, $aC,$aH}
], 3, 3, {1, 1}, {}], {Style["Parameters", 22],
SpanFromLeft}], Alignment -> Center];

(* assemble output *)
Labeled[
Legended[
Show[isoclGraph, critPtsGraph, cstGraph,
Frame -> True,
ImageSize -> Medium,
FrameLabel -> {
{Style[C, Bold, 16], None},
{Style[H, Bold, 16], None}
},
RotateLabel -> False
],
Column[{
LineLegend[
{Directive[{Gray, Thick, Opacity[0.9]}],
Directive[{Lighter[Blue], Thick, Opacity[0.9]}],
Directive[{Brown, Thick, Dashed, Opacity[0.9]}]
},
{Style["Horizontal isoclines", 18],
Style["Vertical isoclines", 18],
Style["Constraint (?)", 18]
}],
Labeled[
PointLegend[
{Directive[{PointSize[Large], Darker[Red]}]},
{Style["Critical points", 18]}
],
critPtsTab
]
}]
],
paramsTab,
Left
]

]

]

]

]
]
]
]
]
]


Are you looking for something like this?

NN = 400; sS = 1/5; sC = 1/7; sH = 1/5; bC = 0.45; bH = 0.4; aC = 0.1; aH = 0.1;

eqH = (bH/NN)*(NN - CC[t] - HH[t])*HH[t] - (sH*aH)*HH[t] == 0;

eqC = (bC/NN)*(NN - CC[t] - HH[t])*CC[t] - (sC + aC)*CC[t] == 0;

eqS = NN - HH[t] - CC[t] - SS[t] == 0

NSolve[{eqH, eqC, eqS}, {CC[t], HH[t], SS[t]}]