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- https://math.libretexts.org/Bookshelves/Calculus/Differential_Calculus_for_the_Life_Sciences_(Edelstein-Keshet)/13%3A_Qualitative_Methods_for_Differential_Equations/13.01%3A_Linear_and_Nonlinear_Differential_Equationsb) From familiarity with power functions (in this case, the functions of N that form the two terms, rN and bN2 ) we can deduce that the second, quadratic term dominates for larger val...b) From familiarity with power functions (in this case, the functions of N that form the two terms, rN and bN2 ) we can deduce that the second, quadratic term dominates for larger values of N, and this means that when the population is crowded, the loss of individuals is greater than the rate of reproduction.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/05%3A_Introduction_to_Differential_EquationsA goal of this chapter is to develop solution techniques for different types of differential equations. As the equations become more complicated, the solution techniques also become more complicated, ...A goal of this chapter is to develop solution techniques for different types of differential equations. As the equations become more complicated, the solution techniques also become more complicated, and in fact an entire course could be dedicated to the study of these equations. In this chapter we study several types of differential equations and their corresponding methods of solution.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01%3A_Population_Dynamics/1.02%3A_The_Logistic_EquationThe exponential growth law for population size is unrealistic over long times. Eventually, growth will be checked by the over-consumption of resources. We assume that the environment has an intrinsic ...The exponential growth law for population size is unrealistic over long times. Eventually, growth will be checked by the over-consumption of resources. We assume that the environment has an intrinsic carrying capacity and populations larger than this size experience heightened death rates.
- https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/07%3A_Nonlinear_Systems/7.02%3A_The_Logistic_EquationIn this section we will explore a simple nonlinear population model. Typically, we want to model the growth of a given population, y(t), and the differential equation governing the growth behavior of ...In this section we will explore a simple nonlinear population model. Typically, we want to model the growth of a given population, y(t), and the differential equation governing the growth behavior of this population is developed in a manner similar to that used previously for mixing problems.
- https://math.libretexts.org/Under_Construction/Purgatory/Book%3A_Active_Calculus_(Boelkins_et_al.)/07%3A_Differential_Equations/7.06%3A_Population_Growth_and_the_Logistic_EquationThe growth of the earth’s population is one of the pressing issues of our time. Will the population continue to grow? Or will it perhaps level off at some point, and if so, when? In this section, we w...The growth of the earth’s population is one of the pressing issues of our time. Will the population continue to grow? Or will it perhaps level off at some point, and if so, when? In this section, we will look at two ways in which we may use differential equations to help us address questions such as these. Before we begin, let’s consider again two important differential equations that we have seen in earlier work this chapter.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/07%3A_Differential_Equations/7.06%3A_Population_Growth_and_the_Logistic_EquationThe growth of the earth’s population is one of the pressing issues of our time. Will the population continue to grow? Or will it perhaps level off at some point, and if so, when? In this section, we w...The growth of the earth’s population is one of the pressing issues of our time. Will the population continue to grow? Or will it perhaps level off at some point, and if so, when? In this section, we will look at two ways in which we may use differential equations to help us address questions such as these. Before we begin, let’s consider again two important differential equations that we have seen in earlier work this chapter.
- https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Integral_Calculus/3%3A_L'Hopital's_Rule_and_Improper_Integrals/3.3%3A_Logistics_EquationsAlternatively, we can assume that the population will never become greater than 50,000 people and that the rate of population growth is proportional to the product of the population and 50,000 minus t...Alternatively, we can assume that the population will never become greater than 50,000 people and that the rate of population growth is proportional to the product of the population and 50,000 minus the population. 1/50000[∫dPP+∫dP50000−P]=kt+Cln(P)−ln(50000−P)=at+blnP50000−P=at+b.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_(Chasnov)/03%3A_First-Order_ODEs/3.04%3A_ApplicationsThe value of the investment at the time t+∆t is then given by S(t+Δt)=S(t)+(rΔt)S(t)+kΔt, where at the end of the time interval ∆t, rΔtS(t) is the...The value of the investment at the time t+∆t is then given by \boldsymbol{\label{eq:1}S(t+\Delta t)=S(t)+(r\Delta t)S(t)+k\Delta t,} where at the end of the time interval ∆t, rΔtS(t) is the amount of interest credited and k∆t is the amount of money deposited (k>0) or withdrawn (k<0).