IGNOU MMT 9 2026 - Do the stability analysis of the following model formulated to study t

Question

Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.
$$\frac{dN_1}{dt} = r_0 N_1 - r_1 C_0 N_1 - b N_1 N_2$
$$\frac{dN_2}{dt} = -d_0 N_2 - d_1 V_0 N_2 + \beta_0 b N_1 N_2$
$$\frac{dC_0}{dt} = k_1 P - g_1 C_0 - m_1 C_0$
$$\frac{dV_0}{dt} = k_2 P - g_2 V_0 - m_2 V_0$
$$\frac{dP}{dt} = Q - hP - kP(N_1 + N_2) + gC_0N_1 + \ell V_0 N_2 \text{.}$

Where all the variables and constants are same as defined in the system (32)-(35) except for the following

$Q = \text{constant input rate}$

$h = \text{decay rate}$

$P(t) = \text{environmental toxicant concentration}$

$k = \text{ingestion rate of toxicant by the populations}$

$g, \ell = \text{return rate of toxicant in the environment after the death of the populations,}$
$\text{assuming that toxicant is non-degradable}$

$Q > 0, h, k, g, \ell \text{ are positive constants.}$

10 Jan 2026

Answer :

Word Count : 281

Consider the autonomous system describing the prey (N_1), predator (N_2), toxicant concentration in prey (C_0), toxicant concentration in predator (V_0), and environmental toxicant (P). Equilibrium points are obtained by setting all time derivatives equal to zero. From [ \frac{dC_0}{dt}=k_1P-(g_1+m_1)C_0=0,\qquad \frac{dV_0}{dt}=k_2P-(g_2+m_2)V_0=0, ] one gets [ C_0^*=\frac{k_1}{g_1+m_1}P^*,\qquad V_0^*=\frac{k_2}{g_2+m_2}P^*. ] From [ \frac{dN_1}{dt}=N_1(r_0-r_1C_0-bN_2)=0,\qquad \frac{dN_2}{dt}=N_2(-d_0-d_1V_0+\beta_0 bN_1)=0, ] a coexistence equilibrium with (N_1^*>0,;N_2^*>0) satisfies [ r_0-r_1C_0^*-bN_2^*=0,\qquad -d_0-d_1V_0^*+\beta_0 bN_1^*=0, ] giving

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IGNOU Solved Assignments Assignments Ignou Projects Project Previous Year Solved Exam Papers Notes Notes Sample Question Papers Sample Question Papers

Customer Orders (in thousands)	Add-delete items (in thousands)	CPU time (in hours)
123.5	2.108	141.5
146.1	9.213	168.9
133.9	1.905	154.8
128.5	0.815	146.5
151.5	1.061	172.8
136.2	8.603	160.1
92.0	1.125	108.5

Return		Probabilities
X	Y	`P_xj=P_yj=P_j`
0.20	0.15	0.30
0.15	0.08	0.25
0.10	0.05	0.15
0.11	0.09	0.25