Lotka–Volterra equations

Here $x$ is the number of prey and $y$ the number of predators, both in thousands. Each changes at a rate per head: the prey grow at $\alpha$ and are eaten in proportion to the predators, and the predators die at $\gamma$ and grow in proportion to the prey. The phase plane follows the point $(x, y)$ from your start, and the time graph below draws both numbers year by year.

The prey's rate $\alpha - \beta y$ is zero on the line $y = \alpha/\beta$, and the predators' rate $\delta x - \gamma$ is zero on the line $x = \gamma/\delta$. These dashed zero-growth lines cross at the balance point $(x^*, y^*) = (\gamma/\delta, \alpha/\beta)$ and cut the plane into four regions. Below $y^*$ the prey increase, and above it they decrease. To the right of $x^*$ the predators increase, and to the left they decrease. So the path goes round counterclockwise. The predators peak when they stop growing, which is when the prey have fallen back to $x^*$, after the prey's own peak. That delay is the lag in the readout.

Going round does not by itself bring the path back; a spiral goes round too. What closes it is a quantity that stays the same along every solution:$$\begin{aligned} V &= \delta x - \gamma \ln x + \beta y - \alpha \ln y, \\ V' &= (\delta x - \gamma)(\alpha - \beta y) + (\beta y - \alpha)(\delta x - \gamma) = 0. \end{aligned}$$Each level of $V$ above its minimum is a closed curve around the balance point, so the path follows it back to its start and repeats. With the ceiling $K$, crowding slows the prey's growth and $V$ is no longer constant; in the ceiling highlight the path spirals in toward the balance point instead of closing.

However big the swings, every full cycle averages exactly the balance point. Dividing the predators' equation by $y$ gives $(\ln y)' = \delta x - \gamma$. Over one cycle $y$ comes back to its start, so this integrates to 0, and the average of $x$ is $\gamma/\delta = x^*$. The prey's equation gives the average of $y$, $\alpha/\beta = y^*$, the same way. Larger swings change the extremes and the period, not the averages.

Fishing both populations continuously at the same rate per head, $\varepsilon$, turns $\alpha$ into $\alpha - \varepsilon$ and $\gamma$ into $\gamma + \varepsilon$. Without the ceiling the balance point moves to $x^* = (\gamma + \varepsilon)/\delta$, $y^* = (\alpha - \varepsilon)/\beta$: more prey on average, fewer predators. The prey's average is the number at which the predators just break even, so it rises when fishing makes the predators die faster.