Existence of solution to dynamical system

Question

How can I prove that there exists a solution to dynamical system presented below?

$$ \dfrac{dT}{dt} = \lambda - \alpha T + rT\bigg(1 - \dfrac{T+I}{T_{max}}\bigg) - kVT \\ \dfrac{dI}{dt} = kVT - \beta I \\ \dfrac{dV}{dt} = N \beta I - \gamma V \\ $$

Answer 1

Your right side is polynomial, thus continuous, so the local existence theorems apply.

As it is also locally Lipschitz, any solution of an initial value problem is unique.