Figure 1

Phase plane for system (2.4). The lines \({\hat{e}=\frac{Q}{\theta}}\) and \(f(x)=\frac{\hat{f}\theta }{Q}\) split the phase plane into three biological significant regions. Region I is defined by \(\hat{e}<\frac{Q}{\theta}\) and \(f(x)<\frac {\hat{f}\theta}{Q}\), where P is neither limiting nor in excess. Region II is defined by \(\hat{e}>\frac{Q}{\theta}\) (always \({f(x)<\frac{\hat{f}\theta}{Q}}\) from Lemma 2.4.1, [21]), where grazer’s growth is limited by a deficiency of P. Region III is defined by \(\hat{e}<\frac{Q}{\theta}\) and \(f(x)>\frac{\hat {f}\theta}{Q}\), where P is in excess and reduces grazer’s growth