Feedback Rates Depending On Current Stock

When the Current Stock Changes Its Own Rate

A stock is an amount that can increase or decrease over time: for example, the volume of water in a tank, the quantity of a stored material, or the number of active users in a system. In a basic stock balance, inflows add to the amount and outflows subtract from it. In many systems, however, a flow itself changes as the stock changes. A larger stock may create more opportunity for removal, or it may alter how strongly replenishment responds.

This is a feedback model: the current amount helps determine its own rate of change. The modeling task is to express that relationship clearly, preserve the direction of each flow, and check that the proposed rate laws make sense over the range where they will be used. The goal here is to construct and interpret these differential models, not to solve them.

Start With a Stock Balance

Let \(S(t)\) denote the amount of a stock at time \(t\). Assume that \(S\) is differentiable on the time interval being modeled. Let \(I(S(t),t)\) be the total inflow rate and \(O(S(t),t)\) the total outflow rate. Both rates are written as nonnegative amounts per unit time; the outflow is subtracted because it removes stock.

The notation makes an important distinction. \(S(t)\) is an amount, while \(\frac{dS}{dt}\) is a rate. A flow law such as \(O(S)=kS\) gives a rate only when the units of \(k\) convert an amount into an amount per time.

Why the Balance Equation Has This Form

The balance equation follows from accounting for the change over a time interval. If the inflow and outflow rates are continuous, the net amount added between times \(t\) and \(t+h\) is the accumulated inflow minus the accumulated outflow. Therefore, for \(h\) small enough to stay in the modeled interval,

Divide by \(h\), then let \(h\) approach zero. By the definition of the derivative and the average-value limit for a continuous integrand, the left side tends to \(S'(t)\), and the right side tends to the instantaneous net flow at time \(t\). Thus

This argument explains why the equation uses the rates at the current stock: the stock during the interval may vary, but in the instantaneous limit the rate is evaluated at its current value \(S(t)\). If there are several inflows or outflows, add all inflow rates and subtract all outflow rates.

Proportional Removal

A common feedback law says that the removal rate is proportional to the amount currently present. If \(k>0\) is the proportionality constant, write

For example, this can describe a fixed fraction of a well-mixed stock being removed per unit time. If \(S\) is measured in liters and time in hours, then \(k\) has units of inverse hours: \((\text{hours})^{-1}\). Multiplying \(k\) by liters gives liters per hour. The parameter \(k\) must be nonnegative for this interpretation as an outflow.

With a constant inflow \(a\), where \(a\geq 0\), the stock balance is

The removal term has a minus sign because it leaves the stock. This model says that the removal rate falls when the stock falls; it does not say that the stock itself falls at a constant rate.

Replenishment That Depends on Stock

Replenishment need not be constant. A rule might specify that the replenishment rate changes in response to the current stock. For instance, a controller may replenish more strongly when a stock is low and less strongly when it is high. Over a specified range, one possible rate law is affine:

where \(a\) is a baseline rate and \(b\) describes how the rate changes with stock. If \(S\) is an amount and time is measured in days, then \(a\) has units of amount per day and \(b\) has units of inverse days. For this to represent a nonnegative replenishment flow, the chosen values and the modeled stock range must satisfy \(a-bS\geq 0\). A formula should not be used beyond the range in which that condition and the modeling assumptions hold.

If a constant withdrawal \(c\geq 0\) also occurs, the balance is \(\frac{dS}{dt}=R(S)-c\). The dependence of \(R\) on \(S\) belongs in the inflow term; it does not change the rule that outflows are subtracted.

Two Stock-Proportional Flows

Sometimes both replenishment and removal depend proportionally on the current stock. A replenishment term proportional to stock can represent a process in which a larger existing stock generates more additions. Whether this interpretation is appropriate depends on the application; a positive proportional term is not automatically a suitable description of every replenishment system.

For a baseline inflow \(a\), stock-proportional replenishment \(gS\), and stock-proportional removal \(kS\), with \(a,g,k\geq 0\), the balance is

Here \(g\) and \(k\) have units of inverse time. Keeping the two flow terms visible before combining them helps make the direction of each process clear. The combined coefficient \(g-k\) indicates the net effect of the two stock-dependent flows, but it does not make either flow disappear from the underlying interpretation.

Check the Law Before Using It

A feedback equation is only as useful as its flow assumptions. Before interpreting its predictions, check the meaning and range of each term.

• Check the units. Every term being added or subtracted in the stock-balance equation must be a rate with the same units.
• Keep the direction clear. Inflows add stock; outflows subtract it. Write the separate flows before simplifying.
• Check nonnegativity. A quantity described as a physical replenishment or removal rate should be nonnegative over its intended range.
• State the valid range. A linear feedback law may be a useful approximation only over a particular range of stock values.
• Check the boundary stock. If the stock cannot be negative, ask whether the proposed flows would still make sense near zero. A model whose outflow exceeds all available stock may need a more careful interpretation in that region.

These checks are part of constructing a model, not optional details added after the calculation. A rate law can be dimensionally consistent and still be inappropriate if its assumptions or range do not match the system.

Common Sign Error: Reversing Inflow and Outflow

Suppose replenishment is \(6\) liters per hour and removal is \(4\) liters per hour. The incorrect calculation \(4-6= -2\) liters per hour treats removal minus replenishment as the stock’s rate of change. It predicts a decrease even though more water enters than leaves.

Interpreting the Model Without Solving It

At any specified stock value, the differential equation gives an instantaneous prediction. If its right-hand side is positive, the stock is increasing at that moment; if it is negative, the stock is decreasing; if it is zero, the modeled inflows and outflows balance at that moment. These statements concern the local rate, not the full future trajectory.

For example, in the reserve model, the net rate at \(30\) kilograms is positive and at \(70\) kilograms it is negative. That comparison shows how the proposed feedback law changes the direction of the predicted motion across those stock values. It does not by itself establish what the stock will be at a later time. Determining the time course would require solving or numerically approximating the differential equation, which is a separate task.

The main modeling habit is to begin with the physical balance, represent each flow as a function of the current stock when appropriate, and then test the result with units, signs, and sample values. This keeps the equation connected to the system it is meant to describe.