F={-r1*x1*x2-r3*x1-r4*x1+r7,
-r1*x1*x2-r2*x2*x3-r5*x2+r8,
-r2*x2*x3+m r3*x1-r6*x3+r9};

sol=NSolve[{F==0/.{m->2,r1->3,r2->6,r3->2,r4->1,r5->1,r6->3,r7->1,r8->1,r9->1},x1>0,x2>0,x3>0},{x1,x2,x3},Reals]

J=D[F/.{m->2,r1->3,r2->6,r3->2,r4->1,r5->1,r6->3,r7->1,r8->1,r9->1},{{x1,x2,x3}}];

Eigenvalues[J/.sol[[1]]]

F={-r1 x1 x2-r5 x1-r6 x1+r11,
-r1 x1 x2-r2 x2 x3-r7 x2+r12,
-r2 x2 x3-r3 x3 x4-r8 x3+r13,
-r3 x3 x4-r4 x4 x5-r9 x4+r14,
-r4 x4 x5+m r5 x1 -r10 x5+r15};

sol=NSolve[{F==0/.{m->2,r1->3,r2->6,r3->2,r4->1,r5->1,r6->3,r7->1,r8->1,r9->1,r10->3,r11->1,r12->1,r13->1,r14->1,r15->1},x1>0,x2>0,x3>0,x4>0,x5>0},{x1,x2,x3,x4,x5},Reals]

J=D[F/.{m->2,r1->3,r2->6,r3->2,r4->1,r5->1,r6->3,r7->1,r8->1,r9->1,r10->3,r11->1,r12->1,r13->1,r14->1,r15->1},{{x1,x2,x3,x4,x5}}];

Eigenvalues[J/.sol[[1]]]

F={-r1 x1 x2-r7 x1-r8 x1+r15,
-r1 x1 x2-r2 x2 x3-r9 x2+r16,
-r2 x2 x3-r3 x3 x4-r10 x3+r17,
-r3 x3 x4-r4 x4 x5-r11 x4+r18,
-r4 x4 x5-r5 x5 x6-r12 x5+r19,
-r5 x5 x6-r6 x6 x7-r13 x6+r20,
-r6 x6 x7+m r7 x1-r14 x7+r21};

sol=NSolve[{F==0/.{m->2,r1->3,r2->6,r3->2,r4->1,r5->1,r6->3,r7->1,r8->1,r9->1,r10->3,r11->1,r12->1,r13->1,r14->1,r15->1,r16->3,r17->1,r18->1,r19->1,r20->1,r21->1},x1>0,x2>0,x3>0,x4>0,x5>0,x6>0,x7>0},{x1,x2,x3,x4,x5,x6,x7},Reals]

J=D[F/.{m->2,r1->3,r2->6,r3->2,r4->1,r5->1,r6->3,r7->1,r8->1,r9->1,r10->3,r11->1,r12->1,r13->1,r14->1,r15->1,r16->3,r17->1,r18->1,r19->1,r20->1,r21->1},{{x1,x2,x3,x4,x5,x6,x7}}];

Eigenvalues[J/.sol[[1]]]