**Formulation** Let us consider a Lotka-Volterra system modelling predator-prey interaction of the form $\begin{align} \frac{d}{dt} N(t) &= N(t)(a-(bP(t)+eN(t))), \\ \frac{d}{dt} P(t) &= P(t)(cN(t)-d), & \forall t \in \mathbb{R}, \end{align}$with given parameters $a,b,c,d>0$ and $e\geq 0$. $N(t)$ stands for the time-dependent prey population and $P(t)$ for the predator population. > [!NOTE] > - If predators are present then there is a negative impact on the growth given by $-N(t)P(t)$. > - In turn, the predator population can only grow if prey is present, which is modelled by the term $cN(t)P(t)$. > - Otherwise, they die with a rate factor of $d$. > - All variables are assumed nondimensional already. > # Stationary Points Stationary Points $(\tilde{N}, \tilde{P})$ have to satisfy $\tilde{N}(a-(b\tilde{P}+e \tilde{N}))=0,\quad \tilde{P}(c \tilde{N}- \tilde{d})=0$ From 2nd equation$\tilde{P}=0 \text{ or } \tilde{N} = d/c.$If $\tilde{P}=0$ then the first equation is satisfied if $\tilde{N}=0$ or $\tilde{N}=\frac{a}{e}$ (if $e \neq 0$). If $\tilde{N}= \frac{d}{c}$ then the first equation is satisfied if $\tilde{P}=\frac{ca-ed}{cb}$. Altogether, the solutions to these equations are $(\tilde{N},\tilde{P})\in\left\{ (0,0), \left( \frac{d}{c}, \frac{ca-ed}{^{3}} \right), \left( \frac{a}{e},0 \right) \right\}.$The last only if $e\neq 0$. # Direction Field & Phase Portrait # Implicit Solution In the case $e = 0$, we can derive implicit solutions. Assuming that $N(t),P(t)>0$ we note that $\begin{align} \end{align}$