Pub Date : 2026-01-27DOI: 10.1016/j.exco.2026.100213
Abujafar Mandal, Sk. Md. Abu Nayeem
<div><div>Let <span><math><mi>G</mi></math></span> be a graph with <span><math><mrow><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow></math></span> being vertices and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> being the corresponding vertex degrees. The Sombor index of <span><math><mi>G</mi></math></span> is denoted by <span><math><mrow><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and is defined by <span><math><mrow><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∼</mo><mi>j</mi></mrow></msub><msqrt><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msqrt></mrow></math></span> where <span><math><mrow><mi>i</mi><mo>∼</mo><mi>j</mi></mrow></math></span> means there exists an edge between the vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. Among many other aspects of this index, the relationship of the Sombor index with adjacency graph energy has been studied by many researchers. In this paper, first, we improve the bounds of the second Zagreb index, which is better than the previous existing bounds, and using that result, we establish a relation between the Sombor index and adjacency graph energy. The extended adjacency matrix of a graph of order <span><math><mi>n</mi></math></span> is the <span><math><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></math></span> real matrix whose <span><math><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math></span>-th element will be <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mfenced><mrow><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></math></span> if <span><math><mrow><mi>i</mi><mo>∼</mo><mi>j</mi></mrow></math></span> and 0 otherwise. The extended energy is the aggregate of the absolute values of the eigenvalues of this matrix. Finally, we establish a relationship between the Sombor index and the extended energy. We find an upper and a lower bound of the Sombor index in terms of the extended energy, order of the graph <span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>, size of the graph <span><math><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></math></span>, maximum degree <span><ma
{"title":"Some new relations between Sombor index and graph energy","authors":"Abujafar Mandal, Sk. Md. Abu Nayeem","doi":"10.1016/j.exco.2026.100213","DOIUrl":"10.1016/j.exco.2026.100213","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with <span><math><mrow><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow></math></span> being vertices and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> being the corresponding vertex degrees. The Sombor index of <span><math><mi>G</mi></math></span> is denoted by <span><math><mrow><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and is defined by <span><math><mrow><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∼</mo><mi>j</mi></mrow></msub><msqrt><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msqrt></mrow></math></span> where <span><math><mrow><mi>i</mi><mo>∼</mo><mi>j</mi></mrow></math></span> means there exists an edge between the vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. Among many other aspects of this index, the relationship of the Sombor index with adjacency graph energy has been studied by many researchers. In this paper, first, we improve the bounds of the second Zagreb index, which is better than the previous existing bounds, and using that result, we establish a relation between the Sombor index and adjacency graph energy. The extended adjacency matrix of a graph of order <span><math><mi>n</mi></math></span> is the <span><math><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></math></span> real matrix whose <span><math><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math></span>-th element will be <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mfenced><mrow><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></math></span> if <span><math><mrow><mi>i</mi><mo>∼</mo><mi>j</mi></mrow></math></span> and 0 otherwise. The extended energy is the aggregate of the absolute values of the eigenvalues of this matrix. Finally, we establish a relationship between the Sombor index and the extended energy. We find an upper and a lower bound of the Sombor index in terms of the extended energy, order of the graph <span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>, size of the graph <span><math><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></math></span>, maximum degree <span><ma","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"9 ","pages":"Article 100213"},"PeriodicalIF":0.0,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-24DOI: 10.1016/j.exco.2025.100212
Shaher Momani , Rabha W. Ibrahim
The -Gamma function provides a two-parameter extension of the classical and -deformed Gamma functions, enabling enhanced modeling of memory and scaling effects in fractional calculus and special functions. This paper presents the definition, key properties, and computational aspects of the -Gamma function Applications in generalized fractional derivatives, integral transforms, and -special polynomials are described, with selected examples and discussion of future directions.
{"title":"Applications of the (q,τ)-Gamma function in fractional calculus and special functions","authors":"Shaher Momani , Rabha W. Ibrahim","doi":"10.1016/j.exco.2025.100212","DOIUrl":"10.1016/j.exco.2025.100212","url":null,"abstract":"<div><div>The <span><math><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span>-Gamma function provides a two-parameter extension of the classical and <span><math><mi>q</mi></math></span>-deformed Gamma functions, enabling enhanced modeling of memory and scaling effects in fractional calculus and special functions. This paper presents the definition, key properties, and computational aspects of the <span><math><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span>-Gamma function Applications in generalized fractional derivatives, integral transforms, and <span><math><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span>-special polynomials are described, with selected examples and discussion of future directions.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"9 ","pages":"Article 100212"},"PeriodicalIF":0.0,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates the existence of almost automorphic, asymptotically almost automorphic, pseudo almost automorphic, and pseudo almost periodic solutions to damped Euler–Lagrange equations. The analysis relies on a variational framework and the continuity properties of Nemytskii operators. Both autonomous and non-autonomous cases, including external forcing terms, are addressed. A central result shows that the set of forcing functions leading to almost automorphic solutions is dense with respect to a suitable norm. The main contribution of this work lies in extending classical existence results for almost periodic solutions to more general classes of functions, thereby emphasizing the robustness and generality of such behaviors in variational systems.
{"title":"Almost automorphic and pseudo almost periodic solutions to damped Euler–Lagrange equations","authors":"Bouadi Abdelkader , Boudjema Souhila , Houmia Anouar","doi":"10.1016/j.exco.2025.100211","DOIUrl":"10.1016/j.exco.2025.100211","url":null,"abstract":"<div><div>This paper investigates the existence of almost automorphic, asymptotically almost automorphic, pseudo almost automorphic, and pseudo almost periodic solutions to damped Euler–Lagrange equations. The analysis relies on a variational framework and the continuity properties of Nemytskii operators. Both autonomous and non-autonomous cases, including external forcing terms, are addressed. A central result shows that the set of forcing functions leading to almost automorphic solutions is dense with respect to a suitable norm. The main contribution of this work lies in extending classical existence results for almost periodic solutions to more general classes of functions, thereby emphasizing the robustness and generality of such behaviors in variational systems.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"9 ","pages":"Article 100211"},"PeriodicalIF":0.0,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145719100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-06DOI: 10.1016/j.exco.2025.100210
W. Dörfler, M. Elasmi, T. Laufer
Physics-Informed Neural Networks (PINNs) face significant challenges when applied to problems governed by the Helmholtz equation, primarily due to its indefinite structure and the well-known spectral bias of neural networks. These issues become particularly pronounced at high wavenumbers. To address this, we propose a novel stabilization strategy that improves the learning capabilities of PINNs by recasting the original boundary value problem into an equivalent one that is more amenable to training. Specifically, we reformulate the model problem by decomposing the total wave into a tapered continuation of the incoming wave and a residual scattered field. This allows for the introduction of an inhomogeneity in the boundary value problem, which improves the flow of information during backpropagation. To illustrate the issue and highlight the effectiveness of our approach, we consider a simplified scattering problem involving an incident wave at the junction of two semi-infinite waveguides, a fundamental setting in communication technology. The numerical results presented are in accordance with the expected behavior, paving the way for a possible alternative approach to predict scattering problems using PINNs.
{"title":"Taper-based scattering formulation of the Helmholtz equation to improve the training process of Physics-Informed Neural Networks","authors":"W. Dörfler, M. Elasmi, T. Laufer","doi":"10.1016/j.exco.2025.100210","DOIUrl":"10.1016/j.exco.2025.100210","url":null,"abstract":"<div><div>Physics-Informed Neural Networks (PINNs) face significant challenges when applied to problems governed by the Helmholtz equation, primarily due to its indefinite structure and the well-known spectral bias of neural networks. These issues become particularly pronounced at high wavenumbers. To address this, we propose a novel stabilization strategy that improves the learning capabilities of PINNs by recasting the original boundary value problem into an equivalent one that is more amenable to training. Specifically, we reformulate the model problem by decomposing the total wave into a tapered continuation of the incoming wave and a residual scattered field. This allows for the introduction of an inhomogeneity in the boundary value problem, which improves the flow of information during backpropagation. To illustrate the issue and highlight the effectiveness of our approach, we consider a simplified scattering problem involving an incident wave at the junction of two semi-infinite waveguides, a fundamental setting in communication technology. The numerical results presented are in accordance with the expected behavior, paving the way for a possible alternative approach to predict scattering problems using PINNs.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"9 ","pages":"Article 100210"},"PeriodicalIF":0.0,"publicationDate":"2025-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145799023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-01DOI: 10.1016/j.exco.2025.100209
Nicholas Johnston, Edmond W.H. Lee, Vehbi E. Paksoy
A semigroup is a chameleon if it is the reduct of both a finitely based involution semigroup and a non-finitely based involution semigroup. Recently, a certain monoid of order six was shown to be a smallest chameleon in the class of monoids. The main objective of the present article is to show that although there exist 2237 monoids of order six up to isomorphism, examples of chameleons different from do not exist. Consequently, up to isomorphism, is the unique smallest example of chameleon in the class of monoids.
{"title":"Uniqueness of the smallest chameleon in the class of monoids","authors":"Nicholas Johnston, Edmond W.H. Lee, Vehbi E. Paksoy","doi":"10.1016/j.exco.2025.100209","DOIUrl":"10.1016/j.exco.2025.100209","url":null,"abstract":"<div><div>A semigroup is a <em>chameleon</em> if it is the reduct of both a finitely based involution semigroup and a non-finitely based involution semigroup. Recently, a certain monoid <span><math><mi>K</mi></math></span> of order six was shown to be a smallest chameleon in the class of monoids. The main objective of the present article is to show that although there exist 2237 monoids of order six up to isomorphism, examples of chameleons different from <span><math><mi>K</mi></math></span> do not exist. Consequently, up to isomorphism, <span><math><mi>K</mi></math></span> is the unique smallest example of chameleon in the class of monoids.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100209"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-07DOI: 10.1016/j.exco.2025.100208
Baoqiang Yan , Donal O’Regan
In this paper, we consider a nonlocal elliptic equation under Neumann boundary conditions with parameter . We show that the number of positive solutions of our problem is equivalent to the number of solutions of an algebraic equation and we also consider the existence and uniqueness of positive solutions of our problem. Some examples are listed to illustrate our theorems.
{"title":"Existence and uniqueness of solutions for nonlocal elliptic problems with Neumann boundary conditions and some examples","authors":"Baoqiang Yan , Donal O’Regan","doi":"10.1016/j.exco.2025.100208","DOIUrl":"10.1016/j.exco.2025.100208","url":null,"abstract":"<div><div>In this paper, we consider a nonlocal elliptic equation under Neumann boundary conditions with parameter <span><math><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math></span>. We show that the number of positive solutions of our problem is equivalent to the number of solutions of an algebraic equation and we also consider the existence and uniqueness of positive solutions of our problem. Some examples are listed to illustrate our theorems.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100208"},"PeriodicalIF":0.0,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145570922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-06DOI: 10.1016/j.exco.2025.100207
Sudhir Kumar, Jyotindra C. Prajapati
Authors have defined absolutely fractional differentiable functions on a closed bounded interval of . Some significant results and equivalent conditions for the function to be absolutely fractional differentiable on a closed bounded interval have been discussed. Several examples of absolutely fractional differentiable functions have been constructed. Counterexamples have also been constructed for some results.
{"title":"Absolutely fractional differentiable functions: Results, examples and counterexamples","authors":"Sudhir Kumar, Jyotindra C. Prajapati","doi":"10.1016/j.exco.2025.100207","DOIUrl":"10.1016/j.exco.2025.100207","url":null,"abstract":"<div><div>Authors have defined absolutely fractional differentiable functions on a closed bounded interval <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span> of <span><math><mi>R</mi></math></span>. Some significant results and equivalent conditions for the function to be absolutely fractional differentiable on a closed bounded interval <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span> have been discussed. Several examples of absolutely fractional differentiable functions have been constructed. Counterexamples have also been constructed for some results.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100207"},"PeriodicalIF":0.0,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-04DOI: 10.1016/j.exco.2025.100206
Andrei Zabolotskii
Tame -tilings are related to Farey graph and friezes; much less is known about wild (not tame) -tilings. In this note, we demonstrate -tilings that are maximally wild: we prove that the maximum wild density of an integer -tiling is and present -tilings over with wild density 1.
{"title":"Wildest SL2-tilings","authors":"Andrei Zabolotskii","doi":"10.1016/j.exco.2025.100206","DOIUrl":"10.1016/j.exco.2025.100206","url":null,"abstract":"<div><div>Tame <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings are related to Farey graph and friezes; much less is known about wild (not tame) <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings. In this note, we demonstrate <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings that are maximally wild: we prove that the maximum wild density of an integer <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tiling is <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> and present <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings over <span><math><mrow><mi>Z</mi><mo>/</mo><mi>N</mi><mi>Z</mi></mrow></math></span> with wild density 1.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100206"},"PeriodicalIF":0.0,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-28DOI: 10.1016/j.exco.2025.100205
Kwadwo Antwi-Fordjour
In this note, we revisit the predator–prey model with a square root functional response and prey refuge, originally analyzed by Chen and Wang (2018). While the original study asserted the global stability of the coexistence equilibrium under specific conditions, we provide an alternative mathematical proof showing that the prey population can go extinct in finite time. We refine the analytical framework, present an exact extinction criterion, and illustrate our findings with a concrete example: fixed parameter values and initial conditions that satisfy the finite-time extinction condition. Both the analytical and numerical results confirm that extinction occurs in finite time, challenging prior long-term coexistence claims. Our findings underscore the importance of incorporating behavioral dynamics, such as prey aggregation in refuge areas, into ecological models.
{"title":"When long-term coexistence fails: Finite time extinction in a nonlinear model","authors":"Kwadwo Antwi-Fordjour","doi":"10.1016/j.exco.2025.100205","DOIUrl":"10.1016/j.exco.2025.100205","url":null,"abstract":"<div><div>In this note, we revisit the predator–prey model with a square root functional response and prey refuge, originally analyzed by Chen and Wang (2018). While the original study asserted the global stability of the coexistence equilibrium under specific conditions, we provide an alternative mathematical proof showing that the prey population can go extinct in finite time. We refine the analytical framework, present an exact extinction criterion, and illustrate our findings with a concrete example: fixed parameter values and initial conditions that satisfy the finite-time extinction condition. Both the analytical and numerical results confirm that extinction occurs in finite time, challenging prior long-term coexistence claims. Our findings underscore the importance of incorporating behavioral dynamics, such as prey aggregation in refuge areas, into ecological models.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100205"},"PeriodicalIF":0.0,"publicationDate":"2025-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-10DOI: 10.1016/j.exco.2025.100204
Shaher Momani , Rabha W. Ibrahim
The biological significance of fractal–fractional calculus resides to analyzing and describing complex biological systems (like the dynamic system of potato spots). These systems frequently have erratic patterns or processes with memory and long-term dependencies. We seek an example to enhance models of dynamic system of potato spots.
{"title":"An example of Disease-induced fractal–fractional dynamic system for potato spots","authors":"Shaher Momani , Rabha W. Ibrahim","doi":"10.1016/j.exco.2025.100204","DOIUrl":"10.1016/j.exco.2025.100204","url":null,"abstract":"<div><div>The biological significance of fractal–fractional calculus resides to analyzing and describing complex biological systems (like the dynamic system of potato spots). These systems frequently have erratic patterns or processes with memory and long-term dependencies. We seek an example to enhance models of dynamic system of potato spots.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100204"},"PeriodicalIF":0.0,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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