The stages module for the L component — stages • ramp.library

A staged aquatic mosquito development model with four larval instars. Population density is structured into eggs ($E$), four larval instars ($L_1$–$L_4$), and pupae ($P$). Density-dependent regulation operates through a weighted mean crowding index $$L = w_1 L_1 + w_2 L_2 + w_3 L_3 + L_4,$$ allowing each instar to contribute differently to crowding. Each larval instar has its own maturation rate, density-dependent maturation delay, density-independent mortality, and density-dependent mortality.

State Variables

E: egg density per habitat
L1: first instar larval density per habitat
L2: second instar larval density per habitat
L3: third instar larval density per habitat
L4: fourth instar larval density per habitat
P: pupal density per habitat

Parameters

psi_E: egg maturation (hatching) rate ($\psi_E$)
phi_E: egg mortality rate ($\phi_E$)
psi1, psi2, psi3, psi4: larval instar maturation rates ($\psi_i$)
xi1, xi2, xi3, xi4: delayed maturation responses to mean crowding ($\xi_i$)
phi1, phi2, phi3, phi4: density-independent larval mortality rates ($\phi_i$)
theta1, theta2, theta3, theta4: density-dependent larval mortality slopes ($\theta_i$)
phi_P: pupal mortality rate ($\phi_P$)
psi_P: pupal maturation (emergence) rate ($\psi_P$)
w1, w2, w3: contributions of $L_1$, $L_2$, and $L_3$ to mean crowding ($w_i$)

Egg Laying

eta: the egg laying rate ($\eta$)

Mean Crowding

$$L = w_1 L_1 + w_2 L_2 + w_3 L_3 + L_4$$

The fourth instar $L_4$ acts as the reference stage ($w_4 = 1$). Setting all $w_i = 1$ recovers unweighted total larval density.

Dynamics

$$dE/dt = \eta - (\psi_E + \phi_E) E$$ $$dL_1/dt = \psi_E E - \psi_1 L_1 e^{-\xi_1 L} - (\phi_1 + \theta_1 L) L_1$$ $$dL_2/dt = \psi_1 L_1 e^{-\xi_1 L} - \psi_2 L_2 e^{-\xi_2 L} - (\phi_2 + \theta_2 L) L_2$$ $$dL_3/dt = \psi_2 L_2 e^{-\xi_2 L} - \psi_3 L_3 e^{-\xi_3 L} - (\phi_3 + \theta_3 L) L_3$$ $$dL_4/dt = \psi_3 L_3 e^{-\xi_3 L} - \psi_4 L_4 e^{-\xi_4 L} - (\phi_4 + \theta_4 L) L_4$$ $$dP/dt = \psi_4 L_4 e^{-\xi_4 L} - (\phi_P + \psi_P) P$$

Emergence

The emergence rate of adult, female mosquitoes from each habitat is: $$\alpha = \psi_P P$$