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The stability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover.

The stability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover.

第一不动点的稳定性非常重要。只有当它是稳定的，非零（非灭绝）物种群体才有可能趋向于消失。因此，即使种群处于初期状态下，其动力系统仍然可能导致两种物种都灭绝。但是，由于第一不动点位于原点处是一个鞍点，因此很不稳定，进而说明模型中的两种物种都灭绝并不容易。（实际上，只有在人为地彻底消灭猎物，导致捕食者饿死之后，才会发生这种情况。如果相反消灭了捕食者，那么在这个简单的模型中，猎物将不受限制地增长。）捕猎模型（捕食者-猎物）中的种群可以无限地接近零，且仍然可以恢复。

第一不动点的稳定性非常重要。只有当它是稳定的，非零（非灭绝）物种群体才有可能趋向于消失。因此，即使种群处于初期状态下，其动力系统仍然可能导致两种物种都灭绝。但是，由于第一不动点位于原点处是一个鞍点，因此很不稳定，进而说明模型中的两个物种都灭绝并没有那么容易。（实际上，只有在人为地彻底消灭猎物，导致捕食者饿死之后，才会发生这种情况。如果相反消灭了捕食者，那么在这个简单的模型中，猎物将不受限制地增长。）捕猎模型（捕食者-猎物）中的种群可以无限地接近零，且仍然可以恢复。