1.3 Mathematical Modeling with Differential Equations

Mathematical modeling with differential equations is a powerful tool for describing real-world phenomena. It allows us to represent complex systems using mathematical language, helping us understand and predict their behavior over time.

In this section, we explore various models, including growth and decay, physical systems, and compartmental models. These applications showcase how differential equations can be used to solve practical problems across different fields of study.

Growth and Decay Models

Exponential Growth and Decay

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• Exponential growth and decay models describe situations where the rate of change of a quantity is proportional to the current amount of that quantity
• The general form of an exponential growth or decay model is $\frac{dy}{dt} = ky$, where $k$ is the growth or decay rate constant
  • If $k > 0$, the model represents exponential growth (population growth, compound interest)
  • If $k < 0$, the model represents exponential decay (radioactive decay, cooling of an object)
• The solution to an exponential growth or decay model is given by $y(t) = y_0e^{kt}$, where $y_0$ is the initial value of the quantity at time $t = 0$

Logistic Growth

• Logistic growth models describe situations where a population grows exponentially until it reaches a carrying capacity, at which point the growth rate slows down and eventually stabilizes
• The logistic growth model is given by $\frac{dy}{dt} = ky(1 - \frac{y}{L})$, where $k$ is the growth rate constant and $L$ is the carrying capacity
  • The term $(1 - \frac{y}{L})$ represents the effect of limited resources on the growth rate as the population approaches the carrying capacity
• The solution to the logistic growth model is $y(t) = \frac{L}{1 + (\frac{L}{y_0} - 1)e^{-kt}}$, where $y_0$ is the initial population size

Population Dynamics

• Population dynamics models describe how populations of different species interact with each other and their environment over time
• These models can include factors such as birth rates, death rates, competition, predation, and resource availability
• Examples of population dynamics models include predator-prey models (Lotka-Volterra equations) and competition models (Competitive Lotka-Volterra equations)

Physical Systems

Newton's Law of Cooling

• Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature
• The model is given by $\frac{dT}{dt} = -k(T - T_a)$, where $T$ is the temperature of the object, $T_a$ is the ambient temperature, and $k$ is the cooling rate constant
• The solution to Newton's law of cooling is $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_0$ is the initial temperature of the object

Mechanical Systems

• Mechanical systems, such as spring-mass systems and pendulums, can be modeled using differential equations
• For example, a spring-mass system can be modeled by $m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)$, where $m$ is the mass, $c$ is the damping coefficient, $k$ is the spring constant, and $F(t)$ is the external force applied to the system
• The solutions to these models describe the motion of the mechanical system over time (oscillations, damping, resonance)

Electrical Circuits

• Electrical circuits can be modeled using differential equations based on Kirchhoff's laws and the relationships between voltage, current, and circuit elements (resistors, capacitors, inductors)
• For example, an RC circuit can be modeled by $RC\frac{dV}{dt} + V = V_s(t)$, where $R$ is the resistance, $C$ is the capacitance, $V$ is the voltage across the capacitor, and $V_s(t)$ is the source voltage
• The solutions to these models describe the behavior of the electrical circuit over time (charging, discharging, transient response)

Compartmental Models

Compartment Models

• Compartment models describe the flow of a substance between different compartments or pools within a system
• These models are often used in pharmacokinetics (drug absorption, distribution, metabolism, and elimination), epidemiology (spread of diseases), and ecology (nutrient cycling)
• The general form of a compartment model is a system of first-order linear differential equations, with each equation representing the rate of change of the amount of substance in a particular compartment
  • For example, a two-compartment model can be represented by $\frac{dA_1}{dt} = -k_{12}A_1 + k_{21}A_2$ and $\frac{dA_2}{dt} = k_{12}A_1 - k_{21}A_2$, where $A_1$ and $A_2$ are the amounts of substance in compartments 1 and 2, and $k_{12}$ and $k_{21}$ are the transfer rate constants between the compartments

Mixing Problems

• Mixing problems involve the mixing of two or more substances in a tank or container, with the goal of determining the concentration of the substances over time
• These problems can be modeled using first-order linear differential equations, with the rate of change of the amount of substance in the tank depending on the inflow and outflow rates and the volume of the tank
• For example, if a tank with volume $V$ contains a mixture with concentration $c(t)$, and a solution with concentration $c_in$ is flowing in at a rate $r_in$ while the mixture is flowing out at a rate $r_out$, the model is given by $V\frac{dc}{dt} = r_in(c_in - c) - r_outc$