The logistic equation in the context of Stieltjes differential and integral equations

. In this paper, we introduce logistic equations with Stieltjes derivatives and provide explicit solution formulas. As an application, we present a population model which involves intraspecific competition, periods of hibernation, as well as seasonal reproductive cycles. We also deal with various forms of Stieltjes integral equations, and find the corresponding logistic equations. We show that our work extends earlier results for dynamic equations on time scales, which served as an inspiration for this paper


Introduction
The logistic equation is ubiquitous in population dynamics.The simplest version of this equation, which was proposed by Pierre-François Verhulst in 1838 (see [2]), has the form where x(t) represents the size of a population at time t, r is the population growth rate, and K is the carrying capacity of the environment, i.e., the maximum population size that can be sustained by the environment.More realistic models assume that r and K are no longer constants and are, in fact, functions of time, which leads to the equation Observe that the logistic equation above is nonlinear; however, dividing Eq. (1.1) by −x(t) 2  and substituting y(t) = 1/x(t), we obtain the nonhomogeneous linear equation y ′ (t) = −r(t)y(t) + r(t) K(t) , whose solution can be obtained using the variation of constants formula.Conversely, starting with the general first-order nonhomogenous linear equation y ′ (t) = p(t)y(t) + f (t) and performing the change of variables x(t) = 1/y(t), we get the logistic equation Thus, the logistic equation can be regarded as an equation for x = 1/y, where y is a (nonzero) solution of a first-order nonhomogenous linear equation.This idea has been employed in [3], which deals with dynamic equations on time scales.Beginning with the first-order nonhomogeneous linear ∆-dynamic equation the authors found that y = 1/u satisfies y ∆ (t) = −(p(t) + f (t)y(t))y(σ(t)), where σ is the forward jump operator.Similarly, starting with the adjoint equation of Eq. (1.2), namely, v ∆ (t) = −p(t)v(σ(t)) + f (t) they found that x = 1/v satisfies x ∆ (t) = (p(t) − f (t)x(σ(t))x(t). (1.4) Hence, it is reasonable to refer to Eq. (1.3) and Eq.(1.4) as to logistic dynamic equations.
In the present paper, we deal with two classes of equations that are more general than dynamic equations, and whose solutions need not be continuous.First, we focus on Stieltjes differential equations, which were introduced and studied e.g. in [7][8][9][10][11][12]14].The concept of the Stieltjes derivative of a function with respect to a left-continuous nondecreasing function g is recalled in Section 2, where we also recall some basic facts on linear Stieltjes differential equations.In Section 3, we show that if u is a (nonzero) solution of the Stieltjes differential equation u ′ g (t) = p(t)u(t) + f (t), (1.5) then y = 1/u is a solution of y ′ g (t)(1 + (p(t) + f (t)y(t))∆ + g(t)) + p(t)y(t) + f (t)y(t) 2 = 0 (1.6) (where ∆ + g(t) = g(t+) − g(t)), or equivalently y ′ g (t) = −(p(t) + f (t)y(t))y(t+).
In Section 4, we show that the logistic ∆-dynamic equations (1.3) and (1.4) represent special cases of the Stieltjes differential equations (1.6) and (1.8) corresponding to a suitable function g.
Theoretical results on logistic differential equations with Stieltjes derivatives are illustrated on an example in Section 5.It describes a simple model of grizzly bears, whose population dynamics involves competition between individuals, periods of hibernation, as well as a seasonal reproductive cycle.
The second part of the paper deals with Stieltjes integral equations.In Section 6, we recall some basic properties of Stieltjes integrals, and present a generalization of the quotient rule; as far as we are aware, this is the first appearance of the quotient rule for Stieltjes integrals in the literature.
In Section 7, we consider three types of linear nonhomogeneous Stieltjes integral equations, namely x(t) = x(t 0 ) + t t 0 (p(s)x(s) + f (s)) dg(s), as well as the pair of dual equations x(t) = x(t 0 ) + t t 0 (p(s)x(s−) + f (s)) dg(s), x(t) = x(t 0 ) + t t 0 (−p(s)x(s+) + f (s)) dg(s), which were recently studied in [13,20].For each of the three equations, we find the corresponding logistic equation satisfied by the function y = 1/x.In comparison with earlier sections, we only assume that g has bounded variation, and do not require left-continuity.Because of this, the corresponding theory covers not only ∆-dynamic equations on time scales (where the corresponding g is left-continuous), but also ∇-dynamic equations (where g is right-continuous).These facts are utilized in Section 8, where we explore the relations between Stieltjes integral versions of the logistic equation and both types of dynamic equations.

Preliminaries on Stieltjes derivatives
Let g : R → R be a nondecreasing and left-continuous function.We shall denote by µ g the Lebesgue-Stieltjes measure associated to g given by see [6,17,18].We will use the term "g-measurable" for a set or function to refer to µ g -measurability in the corresponding sense, and we denote by L 1 g (X, R) the set of Lebesgue-Stieltjes µ g -integrable functions on a g-measurable set X with values in R, whose integral we denote by X f (s) dµ g (s), f ∈ L 1 g (X, R).
Similarly, we will talk about properties holding g-almost everywhere in a set X (shortened to g-a.e. in X), or holding for g-almost all (or simply, g-a.a.) x ∈ X, as a simplified way to express that they hold µ g -almost everywhere in X or for µ g -almost all x ∈ X, respectively.Define Observe that, as pointed out in [9], the set C g has null g-measure and it is open in the usual topology, so it can be uniquely expressed as the countable union of open disjoint intervals, say With this notation, we also define We are now in position to introduce the definition of the Stieltjes derivative of a real-valued function as in [9,11].Definition 2.1.Let f : R → R and t ∈ R \ C g .We define the Stieltjes derivative, or g-derivative, of f at t as follows, provided the corresponding limit exists: In that case, we say that f is g-differentiable at t.

Remark 2.2.
It is important to note that, as explained in [11, Remark 2.2], for t ∈ N g we have as the domain of the quotient function gives the corresponding one-sided limit.Furthermore, since g is a regulated function, it follows that the g-derivative of f at a point t ∈ D g exists if and only if f (t+) exists and, in that case, .
The following result, [11,Proposition 2.5], contains some basic properties of Stieltjes derivatives such as linearity and the product and quotient rules. .
• The product f 1 f 2 is g-differentiable at t and Next, we present the concept of g-absolute continuity introduced in [9], as well as some of its properties.For simplicity, we introduce such concept as part of the following result from 2. The function F satisfies the following conditions: Remark 2.5.Observe that the equality in Eq. (2.2) is, indeed, true for t = a as we are considering the integral over the empty set, which makes the integral null.
We denote by AC g ([a, b], R) the set of g-absolutely continuous functions in [a, b] with values on R. With this notation, we present [9, Proposition 2.4], a result that, in a way, is the converse of Theorem 2.4. , We include the following lemma that, to the best of our knowledge, is not available in the literature.(Only the fact that if f has bounded variation and is bounded away from zero, then 1/ f has bounded variation, is known; see for example [1,Exercise 1.1].)This result shows that, under certain conditions, the multiplicative inverse of a g-absolutely continuous function is also g-absolutely continuous.
A similar reasoning shows that there exist δ a , δ b > 0 such that Note that the family {(t which is compact, so there must be a finite subcover, i.e., there exist t 1 , t 2 , . . ., which shows that 1/ f has bounded variation on [a, b].Finally, assume that f is g-absolutely continuous on [a, b] and let ε > 0. In that case, there As shown in [8, Proposition 5.5], every g-absolutely continuous function is g-continuous according to the following definition from [8].
or continuous with respect to g at t, if for every ε > 0, there exists δ > 0 such that The following result, [8, Proposition 3.2], describes some properties of g-continuous functions, and thus, of g-absolutely continuous functions.
• f is continuous from the left at every t ∈ (a, b]; In particular, g-continuous functions on Finally, we provide some context and information on differential problems with Stieltjes derivatives of the form with t 0 , T, u 0 ∈ R, T > 0, and F : [t 0 , t 0 + T] × R → R. Let us start by clarifying the concept of solution for this type of equations. Definition 2.10.Given τ ∈ (0, T], a solution of Eq. (2.4) on [t 0 , As usual, one of the most important equations in the context of Stieltjes derivatives is the linear differential equation, which has been deeply studied in [7,8,11].In the following result, which can be found in [11,Theorem 3.2] or, more generally, in [7,Theorem 4.3], we introduce the g-exponential map, the unique solution of the homogeneous linear problem.Theorem 2.11.Let p ∈ L 1 g ([t 0 , t 0 + T), R) be such that Then, the set T Finally, in [11,Theorem 3.5] and [7, Proposition 4.12], using the method of variation of constants, the authors obtained the explicit expression of the unique solution of the nonhomogeneous linear equation, which we present in the next theorem.Theorem 2.12.Let p, f ∈ L 1 g ([t 0 , t 0 + T), R) and suppose that (2.5) holds.Then, the function u : [t 0 , t 0 + T] → R defined as u(t) = x 0 exp g (p, t) + exp g (p, t) is the unique solution of u ′ g (t) = p(t)u(t) + f (t), u(t 0 ) = u 0 . (2.7)

The logistic equation in the context of Stieltjes derivatives
In the setting of ordinary differential equations and dynamic equations on time scales, one way of defining the logistic equation is to consider it as the equation for which a change of variables of the form u(t) = (x(t)) −1 yields a linear equation in the corresponding setting.Hence, following the reasonings in [5, Section 2.4], we will obtain the form of the logistic equation in the context of Stieltjes derivatives through the mentioned change of variables.
In what follows we assume that x 0 , t 0 , T ∈ R, T > 0. Let us start by looking at the change of variables above.Suppose u is a function which is a solution of Eq. (2.7).If u(t) = (x(t)) −1 , provided that the corresponding hypotheses are satisfied, we can compute the g-derivative of x using Proposition 2.3.Indeed, clearly, the function 1 is g-differentiable everywhere (except on C g ) and has null g-derivative so, under suitable conditions, (2.1) ensures that At this point, one might be inclined to define the logistic equation with Stieltjes derivatives on [t 0 , t 0 + T] as Eq.(3.1) as it is in the form of Eq. (2.4).However, in doing so, one needs to require that any solution on [t 0 , t 0 + τ) in the sense of Definition 2.10 must also satisfy that 1 + ∆ + g(t)(p(t) + f (t)x(t)) ̸ = 0 for every t ∈ [t 0 , t 0 + τ) ∩ D g .Alternatively, instead of Eq. (3.1), we can consider the more general equation which no longer requires such consideration at the cost of moving away from problems of the form (2.4).Observe that when the Stieltjes derivative coincides with the usual derivative (namely, when g = Id), Eq. (3.2) yields the usual logistic equation.
After these considerations, we define the logistic equation with Stieltjes derivatives as the initial value problem Naturally, since Eq.(3.3) is no longer in the framework of Eq. (2.4), we need to define the concept of solution for this problem in a similar manner.
, is a solution of Eq. (3.3) so, without loss of generality, we shall assume that x 0 ̸ = 0 for the remaining of the section.
Remark 3.3.Remark 2.2 and Proposition 2.9 imply that, for any x ∈ AC g ([t 0 , t 0 + τ], R), Hence, it is clear that x is a solution of Eq. (3.3) if and only if it is a solution of The following result provides an explicit expression for a solution of Eq. (3.3), which is obtained through the solution of the nonhomogeneous linear equation, Eq. (2.6).
Remark 3.5.Let us briefly reflect on the conditions that we are requiring on the map ϕ in the hypotheses of Theorem 3.4.When we ask for ϕ to not vanish on the interval, we are essentially asking for the solution of the nonhomogeneous linear equation to be different from zero on the whole interval, which allows us to properly define the map x on that set.Observe that this condition is also necessary in the ODE setting.Condition (3.5), on the other hand, is a condition that is only relevant in this context (as D g = ∅ when g = Id) and it is equivalent to the requirements for the quotient rule in Proposition 2.3, guaranteeing that the derivative of x exists wherever the derivative of the solution of the nonhomogeneous linear equation exists.
A careful reader might have noticed that in the proof of Theorem 3.4, we obtained (3.9).In other words, we showed that the map x in (3.6) satisfies Eq. (3.1).This might be a bit surprising since Eq.(3.2) is the more general equation.However, as we show in the next result, under the assumption that (2.5) holds, Eq. (3.1) and Eq.(3.2) are equivalent problems in the sense that a solution of one of the problems is a solution of the other one.Proposition 3.6.Let τ ∈ (0, T] and assume that 1 Proof.First, let x : [t 0 , t 0 + τ] → R be such that (3.10) holds.In that case, there exists N ⊂ [t 0 , t 0 + τ) such that µ g (N) = 0 and Let us first show that Observe that this is clear for t ∈ [t 0 , t 0 + τ) \ (N ∪ D g ) as ∆ + g(t) = 0 in that case.Thus, in order to prove (3.13) we need to show that 1 + (p(t) + f (t)x(t))∆ + g(t) ̸ = 0 for all t ∈ [t 0 , t 0 + τ) ∩ D g .Choose an arbitrary t ∈ [t 0 , t 0 + τ) ∩ D g and suppose for the sake of contradiction that 1 + (p(t) + f (t)x(t))∆ + g(t) = 0.Then, since t ∈ D g , we have ∆ + g(t) > 0, so we can write p(t) + f (t)x(t) = −1/∆ + g(t).In that case, (3.12) yields , , which contradicts the assumption of the result.Thus, (3.13) must hold.Now, (3.11) is a direct consequence of (3.12) and (3.13), which finishes the proof of the first part of the result.The second part of the result is trivial since we are implicitly assuming that 1 + (p(t) + f (t)x(t))∆ + g(t) ̸ = 0 for g-a.a.t ∈ [t 0 , t 0 + τ).
In [13, Section 3], the authors introduced the adjoint linear equation of Eq. (2.7) as the equation with y 0 ∈ R and p, f ∈ L 1 g ([t 0 , t 0 + T], R) such that (2.5) holds.Observe that if we define , then Eq. (3.14) can be rewritten as i.e., it can be regarded as a particular case of Eq. (2.7) since [13, Lemma 3.4, statement (iii)] ensures that P, F ∈ L 1 g ([t 0 , t 0 + T], R) and, furthermore, Hence, we have a logistic equation associated with Eq. (3.14), which is determined by Therefore, we define the adjoint logistic equation with Stieltjes derivatives -that is, the logistic equation associated with the adjoint equation (3.14) -as the initial value problem with p, f ∈ L 1 g ([t 0 , t 0 + T), R) such that (3.13) holds.This equation turns out to be a much simpler version of Eq. (3.3).Remark 3.7.In a similar fashion to Remark 3.3, we can see that Eq. (3.3) is equivalent to As a direct consequence of Theorem 3.4, we have the following result providing an explicit solution for (3.16).Theorem 3.8.Let p, f ∈ L 1 g ([t 0 , t 0 + T), R) be such that (2.5) holds and define If there exists τ ∈ (0, T] such that φ(t) ̸ = 0 for t ∈ [t 0 , t 0 + τ] and then, the map y : [t 0 , t 0 + τ] → R defined as for P, F as in (3.15).Let us check that P, F satisfy the conditions of Theorem 3.4.Since we have already shown that P, F ∈ L 1 g ([t 0 , t 0 + T], R) and 1 + P(t)∆ + g(t) ̸ = 0, t ∈ [t 0 , t 0 + T) ∩ D g , all that is left to do is check that the map ϕ in Theorem 3.4 satisfies the corresponding conditions under our hypotheses.
First, observe that for all t ∈ [t 0 , t 0 + T].Now, by definition, where we have used the identity exp g (P, and, using the identity exp g (P, Therefore, ϕ satisfies the conditions in Theorem 3.4 so the map is a solution of Eq. (3.20) and, thus, a solution of Eq. (3.16) as we wanted to show.

Relations between Stieltjes differential equations and dynamic equations
Throughout this section, we assume that the reader is familiar with time scale calculus and dynamic equations.For more information on these topics, see [4,5].
Let T be a time scale, t 0 , t 0 + T ∈ T, T > 0, and denote [t 0 , t 0 + T) T = [t 0 , t 0 + T) ∩ T. The aim of this section is the study of possible relations between the logistic equation with Stieltjes derivatives, Eq. (3.3), and its corresponding counterpart in the context of dynamic equations as described in [5], where x ∆ denotes the ∆-derivative of x and σ : T → T is the forward jump operator.Here, we assume that p and f are defined on the whole [t 0 , t 0 + T) despite the fact that we only need them to be defined on [t 0 , t 0 + T) T for Eq.(4.1).We do this so that we can easily compare Eq. (3.3) and Eq.(4.1).Similarly, we also want to consider the relations that might take place between the adjoint logistic equation, Eq. (3.16), and the corresponding logistic equation that can be deduced from the adjoint linear equation in [5], namely In order to discuss the possible relations between the different logistic equations, we need to consider a context in which we can compare the two types of differential problems.In [8,Section 8.3] and [12,Section 3.3.3],it is shown that equations on time scales can be regarded as a particular case of Stieltjes differential equations when we consider the nondecreasing and left-continuous map g : R → R defined as As pointed out in [8, Section 8.3], g(t) = t for all t ∈ [t 0 , t 0 + T) T , from which it follows that where µ : T → T denotes the graininess function.Theorems 3.49 and 3.51 in [12] establish the mentioned relation between Stieltjes differential problems and dynamic equations on time scales.Furthermore, a closer look at the proofs of these results shows that, in fact, the equivalence is between the Stieltjes derivative and the ∆-derivative.We gathered this information in the following result.Observe that, unlike [12,Theorem 3.49] we do not require continuity from the left at right-scattered points as such condition is always satisfied for ∆-differentiable maps, see [4,Theorem 1.16 Now, given Theorem 4.1, the equivalence between Eq. (3.3) and Eq.(4.1) should be clear.Indeed, if x satisfies Eq. (4.1), [4, Theorem 1.16 (iv)] ensures that Hence, Theorem 4.1 ensures that if x = x • g with g as in (4.3), then for g-a.a.t ∈ [t 0 , t 0 + T), where the last equality follows from (4.4).Hence, x satisfies Eq. (3.3).
The equivalence between (3.16) and (4.2) is done in an analogous manner and we leave it to the reader.

Applications to population models
Impulsive differential equations and equations on time scales can be regarded as particular cases of differential equations with Stieltjes derivatives, see [8,Section 8].This fact was taken into account in [8, Section 9], where the authors showed that some real-life phenomena can be modelled in the context of Stieltjes calculus.Similarly, in [10, Sections 5 and 6], the authors used these relations to show that Stieltjes differential equations can be a better tool than ODEs for population models of species that exhibit very short periods of reproductions or are subject to dormant states in which the population size is unlikely to change in a noticeable manner.With these ideas in mind, and bearing the applications of the usual logistic equation for population models, we want to show that the logistic equations with Stieltjes derivative introduced above can be an adequate tool to describe the behavior of certain species.
During the winter and early spring months, the grizzly bears, like many other bears, enter a stupor stage, during which they reduce their activity as much as possible in order to survive that time of the year.This is possible because, in the months prior to the hibernation stage, they build a layer of fat that they will use to sustain themself during this dormant state.Naturally, this might cause a population of grizzly bears to compete for resources during the months leading to winter.Interestingly, the mating of the grizzly bear occurs during this period of time when the grizzly bear is preparing itself for the winter.However, the development of the embryos goes on hold until the hibernation stage, which eventually leads to the introduction of newborn cubs towards the end of the stupor stage.
We claim that a logistic equation with Stieltjes derivatives can be used to represent the evolution of a population of grizzly bears.To that end, we shall divide years into the four different seasons and we shall assume that one unit of time, denoted by t, represents a full season, which leads to the following classification of time intervals: ], k = 0, 1, 2, . . ., represent the hibernation periods of the population and, for simplicity, we shall assume that the remaining times of the year, namely, (4k + 3  2 , 4k + 4], k = 0, 1, 2, . . ., represent the period of time when the bears prepare for the next winter.
The next step is to select an adequate nondecreasing and left-continuous map g : R → R which reflects the behavior explained above, keeping in mind the information in [10, Section 5]: "[the map g] can be regarded as a time modulator.Discontinuities correspond to sudden changes . . .while constancy intervals correspond to dormant states . . .The greater the slope, the more influence the corresponding times have in the process".Hence, we would like the map g to exhibit the following properties: (a) On intervals of the form (4k, 4k + 3  2 ], k = 0, 1, 2, . . ., the map g should remain constant as during these times, the population is hibernating and, thus, very unlikely to change drastically. (b) At times of the form 4k + 3  2 , k = 0, 1, 2, . . ., the map g should possess a jump discontinuity, representing the introduction of newborns into the population, which we shall assume to happen simultaneously so that they can be represented by impulses.The map g must be continuous everywhere else as there are no other sudden changes in the population.
(c) In the months directly after new individuals are born, g must have a greater slope as newborns are weaker and, therefore, the population size is more volatile.As time progresses, the slope of the function should flatten as new individuals get stronger.In the times immediately prior to the hibernation periods we would want g to have a less steep slope, representing the slowing down of the population as they approach their dormant state.
Since we will be assuming that the evolution of the population starts at t = 0, for simplicity, we shall assume that g is constant on (−∞, 0].Furthermore, given the cyclical nature of the previously described annual phenomena, we will assume that there exists c ∈ R such that Observe that, in particular, this implies that ∆ + g(t) = ∆ + g( 32 ) for all t ∈ D g .An example of a map g : R → R satisfying conditions (a)-(c) and the extra assumptions is and g(t) = g(4) + g(t − 4) for t > 4, see Figure 5.1.We now consider the initial value problem where x 0 > 0 and F : [0, +∞) × R → R is defined as where β > 0 represents the death rate of the population; α > 0, the reproduction rate; and γ > 0 represents the competition strength.Naturally, (5.3)only represents the evolution of a population as long as x(t) ≥ 0, which will be the case for our solution as we will show later.Furthermore, observe that for t ̸ = 4k + 3 2 , k = 0, 1, 2, . . ., and x(t) > 0, we have x ′ g (t) ≤ 0, which shows that the population is bound to decay over time; while x ′ g (t) ≥ 0 for t = 4k + 3 2 , k = 0, 1, 2, . . .and x(t) > 0, which is consistent with the fact that only new members of the population are introduced at such times.Furthermore, during intervals of the form (4k + 3, 4k + 4], k = 0, 1, 2, . . ., the population decays faster as the population increases.The competition term is not present in the equation on the intervals (4k, 4k + 3  2 ], k = 0, 1, 2, . . ., as during hibernation, there is no competition for resources.Of course, given our choice of g, this is not relevant for (4k, 4k + 3 2 ), k = 0, 1, 2, . . ., as they belong to C g and, thus, have measure zero.Nevertheless, for other choices of g this might be relevant.
Consider the maps p, f : [0, +∞) → R defined as ) (5.5) We claim that (5.3) can be rewritten as that is, it is an adjoint logistic equation with Stieltjes derivatives of the form (3.16).Indeed, given that Observe that if t = 4k + 3 2 , k = 0, 1, 2, . . ., then Thus, we can apply Theorem 3.8 on an interval [0, T], T > 0, to obtain a solution of (5.3).To that end, we need to check that p and f in (5.4) and (5.5) satisfy the corresponding hypotheses.Let T > 0. First, observe that p and f are Borel-measurable maps which guarantees that they are g-measurable.Hence, since they are bounded, it follows that p, f ∈ L 1 g ([0, T), R).Furthermore, observe that (2.5) holds since Observe that, in particular, this implies that exp g (p, t) > 0 for all t ∈ [0, T], see Theorem 2.11. Consider Given that f (t) ≥ 0 for all t ∈ [0, T), it follows that φ is nondecreasing.Therefore, In particular, this proves that φ(t) ̸ = 0 on [0, T], which also shows that (3.18) holds since f (t) = 0 for t ∈ D g .Therefore, since the conditions of Theorem 3.8 are satisfied on the whole [0, T], we know that the map is a solution of (5.3).Since exp g (p, t), φ(t) > 0 for t ∈ [0, T], it follows that x(t) > 0 for all t ∈ [0, T] as we claimed before.Given that Theorem 3.8 can be applied for each T > 0, we can obtain a solution on [0, +∞).The following result provides a recursive expression for such map.
In Figure 5.2 we have plotted the solution above for different values of γ.Observe that the population presents the behavior we expected.Indeed, first note that the population remains constant during the hibernation periods.Furthermore, the population decays between generations, and the rate of this decay depends on the competition strength, γ.This can be easily observed by noting that x( 3 2 +) = (1 + α)x 0 = 17 10 in all the graphs in Figure 5.2, however, the population levels at t = 4 are lower for higher values of γ.
In order to study the asymptotic behavior of the solution of (5.3), we will look at the sequences {P k } ∞ k=0 = {x(4k representing the population at the end of the hibernation period and the population after newborns are introduced, respectively.Using the expression for x obtained in Theorem 5.1, we see that {P k } ∞ k=0 satisfies , k = 0, 1, . . ., which, thanks to (5.1), simplifies to with β = β(g(4) − g( 3 2 +)).Furthermore, P k = (1 + α)P k for k = 0, 1, 2, . . . .Let us rewrite Eq. (5.6) in the form where A simple calculation shows that the map H has, in general, two fixed points, namely zero and .
The next result shows that the asymptotic behavior of the sequences {P k } ∞ k=0 and { P k } ∞ k=0 (and therefore of the whole solution x) depends on whether L is positive (i.e., e β < 1 + α) or nonpositive (i.e., e β ≥ 1 + α).
Hence, the sequence {P k } ∞ k=0 is nonincreasing and bounded from below by the unique positive fixed point L, so it is convergent to L.
On the other hand, if x 0 ≤ L then, for k = 0, 1, 2, . . ., In this case, the sequence {P k } ∞ k=0 is nondecreasing and bounded from above by the unique positive fixed point L, so it is convergent to L.
are constant and equal to x 0 and (1 + α)x 0 , respectively.Hence, it follows from Theorem 5.1 that the solution is 4-periodic in this case.
In Figure 5.3 we can observe the different asymptotic behaviors that we can expect from the solution of Eq. (5.3) as described by Theorem 5.2.In particular, we can see that when e β ≥ 1 + α (i.e., when the death rate is high enough) the population is bound to extinction as presented in the first of the graphs.On the other hand, the second and third plot show that if e β < 1 + α (i.e., when the reproduction rate is high enough), we can expect the population to approach an equilibrium state corresponding to a 4-periodic solution shown in the fourth plot.
As a final note, observe that the example here provided is relatively simple.More complicated models can be obtained if we consider the parameters α, β and K to be functions instead, or if we relax the condition (5.1).

Preliminaries on Stieltjes integrals
In the rest of the paper, we focus on the logistic equation in the context of Stieltjes integral equations.We will work with Kurzweil-Stieltjes integrals (also known as Perron-Stieltjes integrals), but we only need some basic properties of these integrals, which are summarized in the present section.A much more comprehensive treatment is available in [15].Alternatively, it would be possible to work with the Young integral, which coincides with the Kurzweil-Stieltjes integral if the integrand and integrator are regulated and one of them has bounded variation (cf.[15,Theorem 6.13.1]).We need the substitution theorem for the Kurzweil-Stieltjes integral (see [15,Theorem 6.6.1]).
Moreover, if f is regulated and g has bounded variation, then h has bounded variation.
Our next goal is to obtain an integral version of the formula We begin with the case when g is a step function.
with the convention that g(a−) = g(a) and g(b+) = g(b).
Proof.The first equality is obvious from the definition of the integral; let us verify the second one.
We now generalize Lemma 6.4 to functions of bounded variation.Proof.It suffices to prove the second equality.Since g has bounded variation, there exist nondecreasing functions g 1 , g 2 : [a, b] → R such that g = g 1 − g 2 .Also, for each i ∈ {1, 2}, there exists a sequence of nondecreasing step functions {g i n } ∞ n=1 which is uniformly convergent to g i .Without loss of generality, we can assume that these sequences are such that for all n ∈ N and i ∈ {1, 2}.Therefore, Consequently, by letting g n = g 1 n − g 2 n for all n ∈ N, we obtain a sequence of finite step functions {g n } ∞ n=1 , which is uniformly convergent to g, and its members have uniformly bounded variation.
Let us again use the convention that g n (a−) = g n (a) and g n (b+) = g n (b) for each n ∈ N. Note that g n (t−) ⇒ g(t−) and g n (t+) ⇒ g(t+) with respect to t ∈ [a, b] (see [15,Lemma 4.2.3]).
Also, there exists an M > 0 such that (apply Lemma 2.7 to f (t) = g(t−)).Hence, for sufficiently large n ∈ N, we have and therefore where the second equality follows from Lemma 6.3 and the third from the uniform convergence theorem for integrals whose integrators have uniformly bounded variation (see [15, Theorem 6.8.8]).
Once we have Theorem 6.5, it is not difficult to obtain the following integral version of the quotient rule, i.e., of the classical formula with the convention that g(a−) = g(a) and g(b+) = g(b).
Proof.It suffices to prove the second equality.Lemma 2.7 implies that 1/g has bounded variation.Using the integration by parts formula in the form presented in [13, Theorem B.6], we get .
The definition of the integral, Theorem 6.5 and Theorem 6.
which completes the proof.Theorem 6.6 is not needed in the rest of this paper, but we hope it might be useful for subsequent research.

Stieltjes-integral versions of the logistic equation
We are now ready to deal with Stieltjes integral equations.In this section, we always assume that g : [a, b] → R has bounded variation (left-continuity is no longer required).We begin with the linear nonhomogeneous equation for all t ∈ [a, b], with the convention that ∆ + g(s) = 0 if s = max(t, t 0 ), and ∆ − g(s) = 0 if s = min(t, t 0 ).
which is the integral version of the Stieltjes differential equation (3.1).Thus, we see that the form of the logistic equation (7.2) is consistent with the form obtained in Section 3. Condition (7.7) corresponds to the earlier condition (3.5), and condition (7.8) reduces to x(t) ̸ = 0.
Note that Eq. (7.1) is a special case of a generalized linear differential equation, whose solution can be explicitly expressed using the variation of constants formula (see e.g.[15,Theorems 7.8.4 and 7.8.5]).Thus, the reciprocal of this solution is a solution of the logistic equation given in Theorem 7.1.
The proof of the second statement is similar.
Remark 7.6.If g is left-continuous, then a function x satisfying (7.9) or (7.10) is also leftcontinuous, i.e., x(t−) = x(t) for all t.In this case, Eq. (7.9) coincides with Eq. (7.1), i.e., we have the following pair of equations: According to Theorem 7.5, if x(t) ̸ = 0 and x(t+) ̸ = 0 for all t, then y = 1/x satisfies one of the following equations: These are integral versions of the Stieltjes differential equations of Eq. (3.4) and Eq.(3.17), respectively, which are equivalent to the two logistic equations presented in Section 3.
Remark 7.7.General solution formulas for Eq.(7.9) and (7.10) were recently published in [20].They resemble the well-known variation of constants formula, and involve solutions of the corresponding homogenenous Stieltjes integral equations.According to Theorem 7.5, explicit solutions of Eq. (7.9) and (7.10) immediately give rise to explicit solution formulas for the two versions of the logistic equation.
Remark 7.8.Theorem 7.5 again requires that x(t) ̸ = 0, x(t−) ̸ = 0, and x(t+) ̸ = 0 for all t ∈ [a, b].The first condition is obviously necessary, for otherwise the definition of y would not make sense.Let us have a closer look on the latter two conditions, trying to avoid x(t−) and x(t+), and express both conditions in terms of x(t).Suppose first that x : [a, b] → R satisfies Eq. (7.9).Using the properties of the Kurzweil-Stieltjes integral and performing similar calculations as in the proof of [13,Lemma 6.5] (which corresponds to the homogenenous case f = 0), we find that x(t−)(1 + p(t)∆g(t)) = x(t)(1 + p(t)∆ + g(t)) − f (t)∆ − g(t), t ∈ (a, t 0 ), (7.17) x(t−) ( Thus, we have shown how to reformulate the conditions x(t+) ̸ = 0 and x(t−) ̸ = 0 in terms of x(t).Note that if g is left-continuous, then the conditions in (7.22) and (7.21) coincide, and the conditions in (7.23) and (7.24) reduce to x(t) ̸ = 0.A similar analysis can be performed for Eq.(7.10).However, it is easier to observe that x : where p(s) = p(−s), f (s) = − f (−s), g(s) = −g(−s).The proof of the fact is similar to the proof in [13,Remark 6.4] (which corresponds to the case f = 0).Notice that we have x(t+) = y(−t−), x(t−) = y(−t+), ∆ + g(t) = ∆ − g(−t), and ∆ − g(t) = ∆ + g(−t).Using these relations, it is clear that the conditions guaranteeing that x(t+) ̸ = 0 and x(t−) ̸ = 0 for Eq.(7.10) can obtained from the conditions derived earlier for Eq.(7.9) by interchanging ∆ + g and ∆ − g, f and − f , and a and b.In this way, we obtain the following counterparts to conditions (7.21)-(7.24):8.3) and (8.4) are integral forms of the two versions of ∆-dynamic logistic equation described in [3] and mentioned in the introduction of the present paper.
To deal with ∇-dynamic equations, we replace g by the integrator h given by Eq. (8.2).The ∇-dynamic equation x(t) = x(t 0 ) + t t 0 (−p(s)x(s) + f (s))∇s is then a special case of Eq. (7.10) with g replaced by h; note that a solution x of Eq. (7.10) satisfies x(s+) = x(s) for all s, because h is right-continuous, and therefore x has the same property.The corresponding logistic equation (7.12) given by Theorem 7.5 is then equivalent to the ∇-dynamic equation y(t) = y(t 0 ) − t t 0 (−p(s) + f (s)y(s))y(ρ(s))∇s, (8.5)where ρ is the backward jump operator.Indeed, if y is a solution of Eq. (7.12), then y(s+) = y(s) (because h is left-continuous).Moreover, h is constant on each interval As far as we are aware, the ∇-dynamic logistic equations (8.5) and (8.6) did not appear in the literature yet.

[ 9 ,Theorem 2 . 4 . 1 .
Proposition 5.4].Let F : [a, b] → R. The following conditions are equivalent: The function F is g-absolutely continuous on [a, b] according to the following definition: for every ε > 0, there exists δ > 0 such that for every open pairwise disjoint family of subintervals [a, b] are continuous on [a, b] when g is continuous on [a, b).

Theorem 6 . 1 .Theorem 6 . 2 .
Assume that h : [a, b] → R is bounded and f , g : [a, b] → R are such that ) f (t) dg(t),whenever either side of the equation exists.The next result describes the properties of indefinite Kurzweil-Stieltjes integrals (see[15,  Corollary 6.5.5]).Let f , g : [a, b] → R be such that g is regulated and b a f dg exists.Then, for every t 0 ∈ [a, b], the function h