[{"_id":"63297","user_id":"31496","language":[{"iso":"eng"}],"publication":"European Journal of Applied Mathematics","type":"journal_article","abstract":[{"lang":"eng","text":"We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:\r\n\\begin{eqnarray*} \\left\\{ \\begin{array}{ll} u_t = u_{xx} - \\chi \\big(\\frac{u}{v} \\partial_x v \\big)_x - uv +B_1(x,t), \\qquad & x\\in \\Omega, \\ t>0, \\\\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \\qquad & x\\in \\Omega, \\ t>0, \\end{array} \\right. \\end{eqnarray*}\r\nwhich was introduced by Short et al. in [40] with \r\n$\\chi=2$\r\n to describe the dynamics of urban crime.In bounded intervals \r\n$\\Omega\\subset\\mathbb{R}$\r\n and with prescribed suitably regular non-negative functions \r\n$B_1$\r\n and \r\n$B_2$\r\n, we first prove the existence of global classical solutions for any choice of \r\n$\\chi>0$\r\n and all reasonably regular non-negative initial data.We next address the issue of determining the qualitative behaviour of solutions under appropriate assumptions on the asymptotic properties of \r\n$B_1$\r\n and \r\n$B_2$\r\n. Indeed, for arbitrary \r\n$\\chi>0$\r\n, we obtain boundedness of the solutions given strict positivity of the average of \r\n$B_2$\r\n over the domain; moreover, it is seen that imposing a mild decay assumption on \r\n$B_1$\r\n implies that u must decay to zero in the long-term limit. Our final result, valid for all \r\n$\\chi\\in\\left(0,\\frac{\\sqrt{6\\sqrt{3}+9}}{2}\\right),$\r\n which contains the relevant value \r\n$\\chi=2$\r\n, states that under the above decay assumption on \r\n$B_1$\r\n, if furthermore \r\n$B_2$\r\n appropriately stabilises to a non-trivial function \r\n$B_{2,\\infty}$\r\n, then (u,v) approaches the limit \r\n$(0,v_\\infty)$\r\n, where \r\n$v_\\infty$\r\n denotes the solution of \r\n\\begin{eqnarray*} \\left\\{ \\begin{array}{l} -\\partial_{xx}v_\\infty + v_\\infty = B_{2,\\infty}, \\qquad x\\in \\Omega, \\\\[1mm] \\partial_x v_{\\infty}=0, \\qquad x\\in\\partial\\Omega. \\end{array} \\right. \\end{eqnarray*}\r\nWe conclude with some numerical simulations exploring possible effects that may arise when considering large values of \r\n$\\chi$\r\n not covered by our qualitative analysis. We observe that when \r\n$\\chi$\r\n increases, solutions may grow substantially on short time intervals, whereas only on large timescales diffusion will dominate and enforce equilibration."}],"status":"public","date_updated":"2025-12-18T20:08:49Z","publisher":"Cambridge University Press (CUP)","volume":33,"author":[{"first_name":"NANCY","full_name":"RODRIGUEZ, NANCY","last_name":"RODRIGUEZ"},{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"date_created":"2025-12-18T19:23:28Z","title":"On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime","doi":"10.1017/s0956792521000279","publication_identifier":{"issn":["0956-7925","1469-4425"]},"publication_status":"published","issue":"5","year":"2021","page":"919-959","intvolume":" 33","citation":{"ama":"RODRIGUEZ N, Winkler M. On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime. European Journal of Applied Mathematics. 2021;33(5):919-959. doi:10.1017/s0956792521000279","chicago":"RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” European Journal of Applied Mathematics 33, no. 5 (2021): 919–59. https://doi.org/10.1017/s0956792521000279.","ieee":"N. RODRIGUEZ and M. Winkler, “On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime,” European Journal of Applied Mathematics, vol. 33, no. 5, pp. 919–959, 2021, doi: 10.1017/s0956792521000279.","apa":"RODRIGUEZ, N., & Winkler, M. (2021). On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime. European Journal of Applied Mathematics, 33(5), 919–959. https://doi.org/10.1017/s0956792521000279","short":"N. RODRIGUEZ, M. Winkler, European Journal of Applied Mathematics 33 (2021) 919–959.","bibtex":"@article{RODRIGUEZ_Winkler_2021, title={On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime}, volume={33}, DOI={10.1017/s0956792521000279}, number={5}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={RODRIGUEZ, NANCY and Winkler, Michael}, year={2021}, pages={919–959} }","mla":"RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” European Journal of Applied Mathematics, vol. 33, no. 5, Cambridge University Press (CUP), 2021, pp. 919–59, doi:10.1017/s0956792521000279."}},{"user_id":"31496","_id":"63318","language":[{"iso":"eng"}],"type":"journal_article","publication":"European Journal of Applied Mathematics","status":"public","abstract":[{"text":"In a planar smoothly bounded domain$\\Omega$, we consider the model for oncolytic virotherapy given by$$\\left\\{ \\begin{array}{l} u_t = \\Delta u - \\nabla \\cdot (u\\nabla v) - uz, \\\\[1mm] v_t = - (u+w)v, \\\\[1mm] w_t = d_w \\Delta w - w + uz, \\\\[1mm] z_t = d_z \\Delta z - z - uz + \\beta w, \\end{array} \\right.$$with positive parameters$ D_w $,$ D_z $and$\\beta$. It is firstly shown that whenever$\\beta \\lt 1$, for any choice of$M \\gt 0$, one can find initial data such that the solution of an associated no-flux initial-boundary value problem, well known to exist globally actually for any choice of$\\beta \\gt 0$, satisfies$$u\\ge M \\qquad \\mbox{in } \\Omega\\times (0,\\infty).$$If$\\beta \\gt 1$, however, then for arbitrary initial data the corresponding is seen to have the property that$$\\liminf_{t\\to\\infty} \\inf_{x\\in\\Omega} u(x,t)\\le \\frac{1}{\\beta-1}.$$This may be interpreted as indicating that$\\beta$plays the role of a critical virus replication rate with regard to efficiency of the considered virotherapy, with corresponding threshold value given by$\\beta = 1$.","lang":"eng"}],"date_created":"2025-12-18T19:33:01Z","author":[{"first_name":"YOUSHAN","last_name":"TAO","full_name":"TAO, YOUSHAN"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"volume":32,"publisher":"Cambridge University Press (CUP)","date_updated":"2025-12-18T20:06:35Z","doi":"10.1017/s0956792520000133","title":"A critical virus production rate for efficiency of oncolytic virotherapy","issue":"2","publication_status":"published","publication_identifier":{"issn":["0956-7925","1469-4425"]},"citation":{"ama":"TAO Y, Winkler M. A critical virus production rate for efficiency of oncolytic virotherapy. European Journal of Applied Mathematics. 2020;32(2):301-316. doi:10.1017/s0956792520000133","ieee":"Y. TAO and M. Winkler, “A critical virus production rate for efficiency of oncolytic virotherapy,” European Journal of Applied Mathematics, vol. 32, no. 2, pp. 301–316, 2020, doi: 10.1017/s0956792520000133.","chicago":"TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for Efficiency of Oncolytic Virotherapy.” European Journal of Applied Mathematics 32, no. 2 (2020): 301–16. https://doi.org/10.1017/s0956792520000133.","short":"Y. TAO, M. Winkler, European Journal of Applied Mathematics 32 (2020) 301–316.","bibtex":"@article{TAO_Winkler_2020, title={A critical virus production rate for efficiency of oncolytic virotherapy}, volume={32}, DOI={10.1017/s0956792520000133}, number={2}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={TAO, YOUSHAN and Winkler, Michael}, year={2020}, pages={301–316} }","mla":"TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for Efficiency of Oncolytic Virotherapy.” European Journal of Applied Mathematics, vol. 32, no. 2, Cambridge University Press (CUP), 2020, pp. 301–16, doi:10.1017/s0956792520000133.","apa":"TAO, Y., & Winkler, M. (2020). A critical virus production rate for efficiency of oncolytic virotherapy. European Journal of Applied Mathematics, 32(2), 301–316. https://doi.org/10.1017/s0956792520000133"},"page":"301-316","intvolume":" 32","year":"2020"},{"abstract":[{"lang":"eng","text":"We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed."}],"status":"public","publication":"European Journal of Applied Mathematics","type":"journal_article","language":[{"iso":"eng"}],"_id":"63314","user_id":"31496","year":"2020","intvolume":" 32","page":"618-651","citation":{"apa":"SURULESCU, C., & Winkler, M. (2020). Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more). European Journal of Applied Mathematics, 32(4), 618–651. https://doi.org/10.1017/s0956792520000236","short":"C. SURULESCU, M. Winkler, European Journal of Applied Mathematics 32 (2020) 618–651.","mla":"SURULESCU, CHRISTINA, and Michael Winkler. “Does Indirectness of Signal Production Reduce the Explosion-Supporting Potential in Chemotaxis–Haptotaxis Systems? Global Classical Solvability in a Class of Models for Cancer Invasion (and More).” European Journal of Applied Mathematics, vol. 32, no. 4, Cambridge University Press (CUP), 2020, pp. 618–51, doi:10.1017/s0956792520000236.","bibtex":"@article{SURULESCU_Winkler_2020, title={Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)}, volume={32}, DOI={10.1017/s0956792520000236}, number={4}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={SURULESCU, CHRISTINA and Winkler, Michael}, year={2020}, pages={618–651} }","ama":"SURULESCU C, Winkler M. Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more). European Journal of Applied Mathematics. 2020;32(4):618-651. doi:10.1017/s0956792520000236","chicago":"SURULESCU, CHRISTINA, and Michael Winkler. “Does Indirectness of Signal Production Reduce the Explosion-Supporting Potential in Chemotaxis–Haptotaxis Systems? Global Classical Solvability in a Class of Models for Cancer Invasion (and More).” European Journal of Applied Mathematics 32, no. 4 (2020): 618–51. https://doi.org/10.1017/s0956792520000236.","ieee":"C. SURULESCU and M. Winkler, “Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more),” European Journal of Applied Mathematics, vol. 32, no. 4, pp. 618–651, 2020, doi: 10.1017/s0956792520000236."},"publication_identifier":{"issn":["0956-7925","1469-4425"]},"publication_status":"published","issue":"4","title":"Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)","doi":"10.1017/s0956792520000236","date_updated":"2025-12-18T20:06:05Z","publisher":"Cambridge University Press (CUP)","volume":32,"author":[{"first_name":"CHRISTINA","full_name":"SURULESCU, CHRISTINA","last_name":"SURULESCU"},{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-18T19:31:21Z"},{"year":"2017","page":"645-684","intvolume":" 29","citation":{"apa":"HILLEN, T., PAINTER, K. J., & Winkler, M. (2017). Global solvability and explicit bounds for non-local adhesion models. European Journal of Applied Mathematics, 29(4), 645–684. https://doi.org/10.1017/s0956792517000328","short":"T. HILLEN, K.J. PAINTER, M. Winkler, European Journal of Applied Mathematics 29 (2017) 645–684.","mla":"HILLEN, T., et al. “Global Solvability and Explicit Bounds for Non-Local Adhesion Models.” European Journal of Applied Mathematics, vol. 29, no. 4, Cambridge University Press (CUP), 2017, pp. 645–84, doi:10.1017/s0956792517000328.","bibtex":"@article{HILLEN_PAINTER_Winkler_2017, title={Global solvability and explicit bounds for non-local adhesion models}, volume={29}, DOI={10.1017/s0956792517000328}, number={4}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={HILLEN, T. and PAINTER, K. J. and Winkler, Michael}, year={2017}, pages={645–684} }","ieee":"T. HILLEN, K. J. PAINTER, and M. Winkler, “Global solvability and explicit bounds for non-local adhesion models,” European Journal of Applied Mathematics, vol. 29, no. 4, pp. 645–684, 2017, doi: 10.1017/s0956792517000328.","chicago":"HILLEN, T., K. J. PAINTER, and Michael Winkler. “Global Solvability and Explicit Bounds for Non-Local Adhesion Models.” European Journal of Applied Mathematics 29, no. 4 (2017): 645–84. https://doi.org/10.1017/s0956792517000328.","ama":"HILLEN T, PAINTER KJ, Winkler M. Global solvability and explicit bounds for non-local adhesion models. European Journal of Applied Mathematics. 2017;29(4):645-684. doi:10.1017/s0956792517000328"},"publication_identifier":{"issn":["0956-7925","1469-4425"]},"publication_status":"published","issue":"4","title":"Global solvability and explicit bounds for non-local adhesion models","doi":"10.1017/s0956792517000328","publisher":"Cambridge University Press (CUP)","date_updated":"2025-12-19T11:03:57Z","volume":29,"date_created":"2025-12-19T11:03:50Z","author":[{"first_name":"T.","last_name":"HILLEN","full_name":"HILLEN, T."},{"first_name":"K. J.","last_name":"PAINTER","full_name":"PAINTER, K. J."},{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"abstract":[{"text":"Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.","lang":"eng"}],"status":"public","publication":"European Journal of Applied Mathematics","type":"journal_article","language":[{"iso":"eng"}],"_id":"63371","user_id":"31496"}]