Solow-Swan Model of Economic Growth with Allee Effect

Akhalaya, Kseniya; Shikhman, Vladimir Article — Published Version Solow-Swan Model of Economic Growth with Allee Effect Journal of Quantitative Economics Suggested Citation: Akhalaya, Kseniya; Shikhman, Vladimir (2025) : Solow-Swan Model of Economic Growth with Allee Effect, Journal of Quantitative Economics, ISSN 2364-1045, Springer India, New Delhi, Vol. 23, Iss. 4, pp. 1259-1278, https://doi.org/10.1007/s40953-025-00468-4 This Version is available at: https://hdl.handle.net/10419/333232 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Journal of Quantitative Economics (2025) 23:1259–1278 https://doi.org/10.1007/s40953-025-00468-4 ORIGINAL ARTICLE Solow‑Swan Model ofEconomic Growth withAllee Effect KseniyaAkhalaya1· VladimirShikhman1 Accepted: 30 June 2025 / Published online: 15 July 2025 © The Author(s) 2025 Abstract In this paper we expand the neoclassical Solow-Swan model of economic growth by introducing population dynamics with Allee effect. Allee effect implies the existence of a threshold for the viability of populations, i.e. a population below this threshold decreases. Above the threshold, the population gradually saturates. We show that the corresponding capital stock per capita may stabilize at two different levels. Both can be expressed in terms of equilibrium points of the standard Solow–Swan model with particular constant population growth rates. Surprisingly enough, the capital stock per capita performs in the long run better if the population becomes extinct, rather then it advances the saturation level. For this conclusion the decrease of population should be relatively moderate compared to the capital depreciation. Keywords Economic growth· Solow-Swan model· Allee effect· Capital stock per capita JEL Classification C62· O40 Introduction In 1956, Solow and Swan proposed a celebrated growth model of an economy (Solow 1956; Swan 1956). The meanwhile classical Solow–Swan model describes how growth of capital and labor force affect the total output of an economy. The output is generated by a neoclassical production function, which exhibits constant returns to scale, positive and diminishing marginal products with respect to capital and labor, and satisfies the so-called Inada conditions. Part of the output will be consumed and the rest will be saved with a constant rate. The gradual decrease in the economic value of the capital stock decreases with the constant rate 𝛿 . The population is assumed to grow exponentially with some rate 𝜆 . By * Vladimir Shikhman [email protected] 1 Department ofMathematics, Chemnitz University ofTechnology, Reichenhainer Str. 41, 09126Chemnitz, Germany 1260 Journal of Quantitative Economics (2025) 23:1259–1278 using these assumptions, a differential equation for the capital stock per capita can be derived. It essentially determines the performance of the economy in the Solow-Swan model. The main result in Solow (1956) says that there is a unique globally asymptotically stable equilibrium point of the underlying differential equation. In economic terms, the economy’s capital stock per capita always stabilizes at a nontrivial level. Since its introduction the Solow-Swan model have been generalized in many directions, see e.g. Barro and Sala-i-Martin (1995). One of the main attempts to make the Solow-Swan model more realistic is to modify the underlying population dynamics. In Scarpello and Ritelli (2003), the authors assumed the population dynamics to follow the logistic law. According to the latter, the population grows, but its growth rate decreases to zero. The corresponding capital stock per capita is shown to stabilize at the equilibrium point of the standard Solow–Swan model with zero population growth rate. In Accinelli and Brida (2006), the same general assumptions on the population size and its growth rate are made. Not surprisingly, the achieved stability results persist. Another generalization, which leads mainly to the qualitatively same findings, has been presented in Accinelli and Brida (2005). Here, the so-called Richards law for population dynamics has been incorporated into the Solow-Swan model. In Guerrini (2006), the population dynamics does not follow a prescribed law. Instead, the population grows, but its growth rate has a limit at infinity. The capital stock per capita is shown to stabilize at the equilibrium point of the standard Solow–Swan model with the constant population growth rate equal to this limit. In this paper, we study the Solow-Swan model with population dynamics underlying the so-called Allee effect. This effect is an important biological phenomenon and was first introduced by Allee in 1930, see e.g. Courchamp etal. (2008). Allee observed during his research on goldfishes that a population with low-density in a certain area leads to the extinction of the species in the long term. Individuals within a population often require the assistance of another individual for more than simple reproductive reasons in order to persist. The most obvious example of this is observed in animals that hunt for prey or defend against predators as a group. In the context of labor, we can think on employees who are relying on the colleagues in their common activities. If some of them depart, the working environment becomes abandoned. This may cause even more people to quit or not to enter the job at all. Allee effect implies the existence of a threshold for the viability of populations, i.e. a population below this threshold decreases and actually becomes extinct over time. However, above the threshold, the population grows in the way similar to the logistic law. Note that the novelty of our approach is two-fold: • Into the same Solow-Swan model we incorporate qualitatively different population dynamics. Depending on the initial value, the population may become extinct or saturate, but its growth rate is always decreasing. • The case of decreasing population is new for the literature on Solow-Swan models. This is if compared to the logistic or Richards law, as well as to implicitly defined population dynamics mentioned above. 1261 Journal of Quantitative Economics (2025) 23:1259–1278 It is worth to mention that the use of Allee effect in the context of Solow-Swan models has been suggested in Khan etal. (2021). There, just a numerical study for the Cobb-Douglas production function with particular attention to fuzzy numbers has been carried out. Let us empirically justify the introduction of the Allee effect in the framework of economic growth models. For that, it is crucial to focus on the novel case of decreasing population below the Allee threshold, i.e. where the population growth rate is negative. For developed countries this is not unusual if one neglects the migration effects, e.g. see Fig.1 for Germany in 1999–2023. Mainly due to the relatively low fertility rates in Germany around 1.5 children per woman, the population would decrease without an additional migration. Our interest here lies in the modeling of exactly this population dynamics and its consequences for the economic growth. Note that anti-migration policies are already implemented in some developed countries, such as e.g. the welfare reduction for immigrants in Denmark Agersnap etal. (2020), or are thought to be implemented by far-right political parties all over Europe Abubakara etal. (2024). In developing countries – we think in first place on China – the rise of population has been bounded for decades by governmental birth restrictions. Consequently, the one-child policy lead to the negative population growth rates, see Fig.2 for China in 1999–2023. Although the Chinese birth policies have been gradually loosened (e.g. in 2013, China allowed couples to have a second child if either parent is an only child, in 2016, it allowed married couples to have two children, and in 2021, it announced support for couples who wish to have a third child), the demographic situation would remain recessionary for decades (Yin 2023). Further, the introduction of Allee effect to human societies can be explained by the depopulation phenomenon from economic geography. There, it is defined as a process in which the population density of an area decreases steadily over time. As briefly summarized in Maya (2024), one distinguishes several reasons for Fig. 1 Population growth rate of Germany in 1999–2023 with and without migration processed by UN, World Population Prospects (2024), OurWorldinData.org/population-growth 1262 Journal of Quantitative Economics (2025) 23:1259–1278 depopulation: "There are social reasons, since depopulation is a driver of inequality due to the decrease in services that declining rural populations receive. There are cultural and emotional reasons that link personal and collective roots to a landscape that is disappearing. There are economic reasons, since there are resources in the territory that are no longer exploited. There are environmental reasons since the ability to control certain impacts on the environment, such as forest fires, is reduced. There are geopolitical reasons, since society loses sovereignty over depopulated territories, opening an opportunity window to other power networks, such as crime or other political agents, to come to control them." Overall, the more people abandon an area the more of their fellows decide to do the same. This positive feedback, which accounts for depopulation, motivates our use of Allee effect for modeling the population dynamics here. Additionally, we refer to Merino and Prats (2020) for the impact of economic and governmental factors on depopulation, as well as to Kulcsár (2016) for historical examples at both the national and local levels. Our findings on the Solow-Swan model with Allee effect are as follows. Depending on the population initial value, the capital stock per capita stabilizes at two different levels. First, if the population dynamics starts below the Allee threshold, the capital stock per capita is shown to stabilize at the equilibrium point of the standard Solow–Swan model with a particular constant population growth rate. The latter constant equals −𝜆 , the negative of the intrinsic population growth rate. More interestingly, the global asymptotic stability is guaranteed iff 𝛿>𝜆 . In economic terms, the capital stock per capita may stabilize even if the population becomes extinct. However, this happens just if the decrease of population is relatively moderate, i.e. the intrinsic population growth rate 𝜆 does not exceed the rate of capital depreciation 𝛿 . Second, if the population dynamics starts above the Allee threshold, the capital stock per capita is shown to stabilize at the equilibrium point of the standard Solow–Swan model with zero population growth rate. This more or less typical behavior is in accordance with Fig. 2 Population growth rate of China in 1999–2023 with and without migration processed by UN, World Population Prospects (2024), OurWorldinData.org/population-growth 1263 Journal of Quantitative Economics (2025) 23:1259–1278 the results in Accinelli and Brida (2006) and Guerrini (2006). The comparison of both equilibrium points of the Solow-Swan model with Allee effect leads to a surprising conclusion. Namely, the corresponding capital stock per capita performs in the long run better if the population becomes extinct, rather then it advances the saturation level. Finally, it is worth to mention that our model remains neoclassical. Here, the assumption on the population dynamics via Allee effect is exogenous. This is in strong contrast with overlapping-generation models, see e.g. Galor and Weil (2000); Yin (2023), where the growth of population is explained endogenously. In Galor and Weil (2000), the transition from Mathusian stagnation to the demographic transition and beyond is considered. The authors argue that in what they call the Modern Growth Regime by then around 2000 "many rich countries have population growth rates near zero", and further – "over the next several decades much of Western Europe is forecast to have negative population growth." Our paper assumes the Allee effect in order to analyze the economic growth corresponding to the latter demographic trend of diminishing population. By doing so, we enlarge the scope of the analysis in Galor and Weil (2000). The paper is organized as follows. In Sect. 2, we introduce the Solow-Swan model with Allee effect. Section3 is devoted to the stability analysis of its equilibrium points. In Sect.4, we discuss the Solow-Swan model with Allee effect for the Cobb-Douglas production function. Model Description We consider a closed economy consisting of a single good Y(t), the community’s real income. The output Y(t) is the measure term for economy growth and will be generated by capital K(t) and labor force L(t). We assume the neoclassical production function F ∶ℝ 2 + →ℝ + with the following properties: (P1) The function F exhibits constant returns to scale, i.e. (P2) The function F exhibits positive and diminishing marginal products with respect to each input K>0 and L>0 : (P3) The function F fulfils the Inada conditions (Inada 1963): Note that the neoclassical properties of the production function imply that the two inputs, K and L, are each essential for production, see e.g. Färe and Primont (2002): Let the level of technological possibilities T>0 be constant. We have F(aK,aL)=aF(K,L)for all a>0. 𝜕 F 𝜕K >0, 𝜕F 𝜕L >0 and 𝜕 2 F 𝜕K 2<0, 𝜕 2 F 𝜕L 2< 0. lim K → 0 𝜕F 𝜕K =lim L → 0 𝜕F 𝜕L =∞, lim K → ∞ 𝜕F 𝜕K =lim L → ∞ 𝜕F 𝜕L = 0. F(K,0)=F(0, L)=0. 1264 Journal of Quantitative Economics (2025) 23:1259–1278 Part of each instant’s output will be consumed and the rest with constant rate s∈[0, 1] saved or invested. We also assume the gradual decrease in the economic value of the capital stock with the rate 𝛿>0 . The following increase of capital stock is thus given by We intend to investigate the per capita income which can be simplified by using (P1): where the capital stock per unit of effective labor is set as Per capita considerations help to measure a sectors’ average incomes and improve the comparison of different economies. The main interest of the model is the dynamics of k(t). Taking derivatives with respect to time t, we obtain where is the growth rate of population. The differential equation in (4) is the economic key of the Solow-Swan model. Now, we turn our attention to the assumption on the population dynamics. In the canonical Solow-Swan model (Solow 1956; Swan 1956), the growth rate n(t) is assumed to be constant, i.e. where 𝜆>0 is an intrinsic growth rate. Hence, the population size yields exponential behavior over time and for any initial level L0>0 , at time t the level of labor force is This behavior represents a useful model for simple populations over relatively short periods of time without accommodation of growth reductions due to limited resources. In Scarpello and Ritelli (2003), a modification to the canonical SolowSwan model was proposed by assuming a logistic population growth. This is an (1) Y(t)=T ⋅ F(K(t),L(t)). (2) K�(t)=s ⋅ Y(t)−𝛿 ⋅ K(t). (3) y(t)=Y(t) L ( t )=T⋅F ( K(t) L ( t ),1 ) =T⋅F(k(t),1) , k (t)= K(t) L(t) . (4) k �(t)= K�(t) ⋅ L(t)−K(t) ⋅ L�(t) L 2 (t) =s⋅y(t)−(𝛿+n(t))⋅k(t) , n (t)=L � (t ) L(t) L�(t)=𝜆 ⋅ L(t), L (t)=L 0 e 𝜆t. 1265 Journal of Quantitative Economics (2025) 23:1259–1278 alternative model, which states that a stable population would have a characteristic saturation level M>0 . The corresponding equation is the well-known logistic differential equation which can be solved by separation of variables. The analytical solution is given by Under this assumption on population growth the economy has a unique non-trivial equilibrium point in terms of the capital stock per capita. In this paper, we go a step further and extend the logistic population growth with the Allee effect, see e.g. Courchamp etal. (2008). A mathematical representation of the Allee effect for population size is given by The parameter N is the critical value or the so-called Allee threshold. We couple (7) with the initial condition L(t0)=L0>0 to form the initial value problem. Unfortunately, the unique solution L(t) of (7) cannot be given explicitly in the general case. Therefore, we look at its qualitative behavior depending on the initial value L0 , see Fig.3 with N=1 and M=2 . We observe asymptotically stable equilibrium points M and 0 as well as an unstable equilibrium point N. Let us now assume the population law with Allee effect from (7) for the SolowSwan model (4). We see that the economy of this modified Solow-Swan model is described by the differential equation (5) L �(t)=𝜆⋅L(t)⋅ ( 1−L(t) M ), (6) L (t)= M 1+e−𝜆t ( M L0 −1 ). (7) L �(t)=𝜆⋅L(t)⋅ ( 1−L(t) M ) ⋅ ( L(t) N−1 ). Fig. 3 Population growth (7) with Allee effect 1266 Journal of Quantitative Economics (2025) 23:1259–1278 where In what follows, we shall study the corresponding initial value problem of the Solow-Swan model with Allee effect for given capital stock per capita k(t0)=k0>0 . Stability Analysis Let us show that the differential equation (8) is uniquely solvable. For that, we first study the behavior of the population growth rate n(t) depending on the initial value L0>0 . From we obtain: By additionally using the properties of L(t) and L�(t) , we may distinguish the following cases for the population growth rate: (1) If L0∈(0, N) , then it holds: (2a) If L 0∈ ( N, N+M 2] , then it holds: where  t is the unique solution of L (t)= N+M 2 . (2b) If L 0∈ (N+M 2 ,M ) , then it holds: (3) If L0∈(M,∞) , then it holds: (8) k�(t)=s ⋅ T ⋅ F(k(t) ,1 )−(𝛿+n(t)) ⋅ k(t), n (t)=L � (t) L(t)=𝜆⋅ ( 1−L(t) M ) ⋅ ( L(t) N−1 ). n (t)=𝜆⋅ ( 1−L(t) M ) ⋅ ( L(t) N−1 ) n �(t)=𝜆⋅L�(t)⋅ ( 1 M+1 N−2L(t) MN ). n (t)<0, n � (t)<0, lim t→∞ n(t)=−𝜆 . n (t)>0, n�(t) ⎧ ⎪ ⎨ ⎪ ⎩ >0 if t< t =0 if t= t <0 if t> t , lim t→∞ n(t)= 0, n (t)>0, n � (t)<0, lim t→∞ n(t)= 0. 1273 Journal of Quantitative Economics (2025) 23:1259–1278 (ii) We know that for L0∈(M,∞) the population growth rate has the following properties, see case (3) from above: The proof of the assertion here goes along the same lines as in Theorem4. We just add n( t) to the depreciation rate and subtract it from the population growth rate: Here,  t is taken sufficiently large to guarantee that 𝛿+n( t)>0 . Clearly, (8) can be equivalently written as (11), where (12) holds with Since the shifted depreciation rate  𝛿 remains positive, the stability analysis from Theorem3 applies. It says that the capital stock per capita for (11) stabilizes at  k∗ −n( t) . Recalling  𝛿=𝛿+n( t) , this equilibrium point corresponds to k∗ 0 , and the assertion follows. (iii) We know that for L0∈{N,M} the population growth rate vanishes, see case (4) from above: Then, (8) becomes the standard Solow-Swan model with zero population growth rate. Thus, the assertion trivially follows. ◻ From Theorems4 and5 the instability of the remaining equilibrium points of (9) follows. Corollary 1 (Instability) The equilibrium points (k∗ 0,N) , (0, M), (0,N), and (0,0) of the Solow-Swan model (9) with Allee effect are unstable. Let us compare the performance of the capital stock per capita in the long run depending on the population growth. Remark 2 (Comparison of equilibria) We assume that 𝛿>𝜆 . Then, according to Theorems4 and5, the Solow-Swan model (8) with Allee effect has two globally asymptotically stable equilibrium points k∗ −𝜆 and k∗ 0 . Since k∗ n is strictly decreasing with respect to n∈ (−𝛿,∞) , we have: From the economic point of view, the capital stock per capita performs in the long run better if the population becomes extinct, rather then it advances the saturation level. n (t)<0, n � (t)>0, lim t→∞ n(t)= 0.  𝛿=𝛿+n(  t) , n(t)=n(t)−n(  t). n=−n( t),n∞=−n( t). n(t) ≡ 0. k∗ −𝜆>k∗ 0. 1274 Journal of Quantitative Economics (2025) 23:1259–1278 Let us establish bounds on the capital stock per capita in terms of the standard Solow-Swan model with constant population growth rates. Remark 3 (Bounds on capital stock per capita) We assume that 𝛿>𝜆 . From the proof of Theorem4 we see that in case of L0∈(0, N) it holds for all t≥t0 : Analogously, from the proof of Theorem 5 we see that in case of L0∈(N,M) it holds for all sufficiently large t≥t0 : In case of L0∈(M,∞) it holds for all t≥t0 : In economic terms, the capital stock per capita is bounded by that of the standard Solow-Swan model with constant population growth rates −𝜆 , 0, and n(t0) , respectively. Discussion forCobb‑Douglas Production Function In this section, we verify the results from Sect.3 on an illustrative example. For that, the two-dimensional system of differential equation (9) will be solved numerically for a given production function satisfying (P1)-(P3). We choose the well-known Cobb-Douglas production function: with 𝛼∈(0, 1) . The Solow-Swan model (9) with Allee effect then becomes Additionally, we make a comparison of (18) with the Solow-Swan model which assumes the population dynamics according to the logistic law, cf. Guerrini (2006); Scarpello and Ritelli (2003): k n(t 0 )( t,t 0 ,k 0)≤ k ( t,t 0 ,k 0)≤ k −𝜆( t,t 0 ,k 0) . k n(t 0 ) (t,t 0 ,k 0 )≤k(t,t 0 ,k 0 )≤k 0 (t,t 0 ,k 0 ). k 0 (t,t 0 ,k 0 ) ≤ k(t,t 0 ,k 0 ) ≤ kn ( t0 ) (t,t 0 ,k 0 ). F(K,L)=K𝛼 ⋅ L1−𝛼 (18) ⎧ ⎪ ⎨ ⎪ ⎩ k�=s⋅T⋅k𝛼− � 𝛿+𝜆⋅ � 1−L M � ⋅ � L N−1 �� ⋅k , L�=𝜆⋅L⋅ � 1−L M � ⋅ � L N−1 � . (19) ⎧ ⎪ ⎨ ⎪ ⎩ k�=s⋅T⋅k𝛼− � 𝛿+𝜆⋅ � 1− L M �� ⋅k , L�=𝜆⋅L⋅ � 1− L M � . 1275 Journal of Quantitative Economics (2025) 23:1259–1278 Assuming 𝛿>𝜆 , we get the equilibrium points of the capital stock per capita for (18) in explicit form, cf. Theorems4 and5: The equilibrium point of the capital stock per capita for (19) is just k∗ 0 , see (Guerrini 2006; Scarpello and Ritelli 2003). We aim to investigate the dependence of the solutions of (18) and (19) on the population initial value. The initial value L0 can be below the Allee threshold N, between the Allee threshold N and the saturation level M, and above the saturation level M. We refer to Figs.4-6 for these three cases. Here, the solid line corresponds to the Solow-Swan model (18) with Allee effect, and the dotted line to the Solow-Swan model (19) with logistic law. The parameters related to capital are chosen as follows: The parameters determining the population dynamics are set as The main difference between assuming Allee effect or logistic law in the SolowSwan model is if starting below the Allee threshold N, see Fig.4. It turns out that then the capital stock per capita stabilizes at a higher level k∗ −𝜆 for the Allee effect in comparison to the lower level k∗ 0 for the logistic law. If starting above the Allee threshold N, see Figs.5 and6, the behavior of the capital stock per capita for the Allee effect is as expected similar to that for the logistic law. In order to illustrate the blow-up effect, which occurs if the stability condition 𝛿>𝜆 is violated, let us take k ∗ −𝜆= ( s⋅T 𝛿−𝜆) 1 1−𝛼,k∗ 0= ( s⋅T 𝛿) 1 1−𝛼 . 𝛼=0.3, s=0.025, T=1, 𝛿=0.075. 𝜆=0.025, N=1, M=2. 𝜆=0.085. Fig. 4 Case L0∈(0, N) 1276 Journal of Quantitative Economics (2025) 23:1259–1278 If starting below the Allee threshold N, the capital stock per capita explodes in this case, see Fig.7. This is in contrast with the logistic law, where the corresponding capital stock per capita stabilizes at k∗ 0 once again. Fig. 5 Case L0∈(N,M) Fig. 6 Case L0∈(M,∞) 1277 Journal of Quantitative Economics (2025) 23:1259–1278 Acknowledgements The authors would like to thank the anonymous referees for suggesting valuable improvements to the paper. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/. References Solow, R. M. 1956. A contribution to the theory of economic growth. The Quarterly Journal of Economics 70:65–94. https:// doi. org/ 10. 2307/ 18845 13. Swan, T. W. 1956. Economic growth and capital accumulation. Economic Record 32:334–361. https:// doi. org/ 10. 1111/j. 14754932. 1956. tb004 34.x. Barro, R.J., and Sala-i-Martin, X. 1995. Economic Growth. McGraw-Hill, New York https:// mitpr ess. mit. edu/ 97802 62025 539/ econo micgrowth/ Scarpello, G. M., and D. Ritelli. 2003. The Solow model improved through the logistic manpower growth law. Annali dell’Universitá di Ferrara 49:73–83. https:// doi. org/ 10. 1007/ BF028 44911. Accinelli, E., and Brida, J.G. 2006. Re-formulation of the Solow economic growth model with the Richards population growth law. In: Proceedings of the MASSEE International Congress on Mathematics, https:// ssrn. com/ abstr act= 874971 Accinelli, E., and Brida, J.G. 2005. Economic growth and population models. GE, Growth, Math methods 0508006 https:// ssrn. com/ abstr act= 874971 Guerrini, L. 2006. The Solow-Swan model with a bounded population growth rate. J. Math. Econ. 42:14– 21. https:// doi. org/ 10. 1016/j. jmate co. 2005. 05. 001. Courchamp, F., Berec, J., and Gascoigne, J. 2008. Allee Effects in Ecology and Conservation. Oxford University Press, New York https:// acade mic. oup. com/ book/ 6171 Fig. 7 Blow-up for 𝛿<𝜆 1278 Journal of Quantitative Economics (2025) 23:1259–1278 Khan, N. A., O. A. Razzaq, A. A. F.Riaz, and N. Senue. 2021. Dynamics of fractional order nonlinear system: A realistic perception with neutrosophic fuzzy number and Allee effect. J. Adv. Res. 32:109–118. https:// doi. org/ 10. 1016/j. jare. 2020. 11. 015. Agersnap, O., A. Jensen, and H. Kleven. 2020. The welfare magnet hypothesis: Evidence from an immigrant welfare scheme in Denmark. American Economic Review: Insights 2:527–542. https:// doi. org/ 10. 1257/ aeri. 20190 510. Abubakara, I., R. Langellaa, and N. Medab. 2024. Europe’s anti-migration policies: the need to reverse a trajectory towards death, despair, and destitution. The Lancet 403:2465–2467. https:// doi. org/ 10. 1016/ S01406736(24) 00922-X. Yin, Y. 2023. China’s demographic transition: A quantitative analysis. Eur. Econ. Rev. 160 : 104591. https:// doi. org/ 10. 1016/j. euroe corev. 2023. 104591. Maya, F.L. Ecology and rural depopulation. 19/07/2024 CREAF Opinion. https:// www. creaf. cat/ en/ artic les/ ecolo gyandruraldepop ulati on Merino, F., and M. A. Prats. 2020. Why do some areas depopulate? The role of economic factors and local governments. Cities 97 : 102506. https:// doi. org/ 10. 1016/j. cities. 2019. 102506. , L. J. 2016. Depopulation and its challenges for development: An international comparison. Journal of Population Problems 72:323–349. https:// www. ipss. go. jp/ publi cation/ e/ jinko mon/ pdf/ 20238 603. pdf Galor, O., and D. N. Weil. 2000. Population, technology, and growth: from Malthusian stagnation to the demographic transition and beyond. American Economic Review 90:806–828. https:// doi. org/ 10. 1257/ aer. 90.4. 806. Inada, K.-I. 1963. On a two-sector model of economic growth: comments and a generalization. Rev. Econ. Stud. 30:119–127. https:// doi. org/ 10. 2307/ 22958 09. Färe, R., and D. Primont. 2002. Inada conditions and the law of diminishing rreturns. International Journal of Business and Economics 1:1–8. https:// ideas. repec. org/a/ ijb/ journl/ v1y20 02i1p18. html Simonovits, A. 2000. Mathematical Methods in Dynamic Economics. Macmillan Press, London https:// doi. org/ 10. 1057/ 97802 30513 532 Waltma, P. 2004. A Second Course in Elementary Differential Equations. Dover Publications, Mineola, New York https:// doi. org/ 10. 1016/ C2013-011666-2 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.