The gravity model (GM) is probably the most known approach to describe empirically the commuter fluxes between cities [12]. It is based on a phenomenological analogy with gravity, assuming that the interaction between two regions or cities depends in an inverse proportionality with the distance raised at a positive power and in direct proportionality with the size of the two regions/cities. Contrary with what is usually believed, GM is not only a simple analogy there are also theoretical arguments in favour of it. The oldest one is probably the one using the maximal entropy hypothesis [13, 14]. Other successful attempts are based on the principle of utility maximization in economics. Both deterministic [15, 16] and random utility theories [17] were considered.

In the most general form, the number of commuters \(f_{i}(j)\) between cities i and j is written as:

$$ f_{i}(j)=F(W_{i}) \frac{(W_{j})^{\alpha}}{(r_{i,j})^{\beta}}. $$ (1)

We denoted here by \(W_{i}\) the population of the settlement i and by \(r_{i,j}\) the distance between settlements i and j. \(F(x)\) is an arbitrary monotonically increasing kernel function, and α and β are fitting exponents. From the \(f_{i}(j)\) data one can also compute the \(P^{i}_{>}(W)_{\mathrm{GM}}\) probability, that a worker living in location i commutes to a location that is outside of a disk containing a population W and centred at its home:

$$ P^{i}_{>}(W)_{\mathrm{GM}}=1-\frac{\sum_{j

e i}^{(w_{i}[j]< W)} f_{i}(j)}{\sum_{j} f_{i}(j)}=1-\frac{\sum_{j

e i}^{(w_{i}[j]< W)} \frac{(W_{j})^{\alpha }}{(r_{i,j})^{\beta}}}{\sum_{j} \frac{(W_{j})^{\alpha}}{(r_{i,j})^{\beta}}}, $$ (2)

which is independent of the \(F(x)\) kernel function. We denoted by \(w_{i}[j]\) the total population inside a disk centred at location i and reaching to location j. Now, the \(P_{>}(W)_{\mathrm{GM}}\) probability that commuters travel to work at a distance where they pass a disk with population W is:

$$ P_{>}(W)_{\mathrm{GM}}= \bigl\langle P^{i}_{>}(W)_{\mathrm{GM}} \bigr\rangle _{i} . $$ (3)

The GM model in such sense is a two parameter model and one has to determine the best α and β exponent values.

The original radiation model (RM) [5] is based on the simple assumption that jobseekers are optimizing their income by accepting the closest job offer that offers a better salary than the one available at their current address. Assuming a \(p_{\le}(z)\) distribution function for the incomes in the studied society the probability \(P_{>}(z|n)\) that a person with income z refuses the closest n jobs is:

$$ P_{>}(z|n)= \bigl[p_{\le}(z) \bigr]^{n}. $$ (4)

By using the probability density function for incomes, the probability of not accepting the closest n jobs, \(P_{>}(n)\), can be calculated as:

$$\begin{aligned} P_{>}(n)&= \int_{0}^{\infty}P_{>}(z|n) p(z) \,dz= \int_{0}^{\infty }P_{>}(z|n) \frac{\partial p_{\le}(z)}{\partial z} \,dz \\ & = \int_{0}^{1} \bigl[p_{\le}(z) \bigr]^{n} \,dp_{\le}(z)=\frac{1}{n+1}. \end{aligned}$$ (5)

Accepting now the hypothesis that the number of job openings in a territory is proportional with the W population (\(n=\mu W\)), the radiation model predicts the probability that a person commutes to a location that is outside of a disk centered on its current location and containing a population W:

$$ P_{>}(W)_{\mathrm{RM}}=\frac{1}{\mu W+1}. $$ (6)

It is interesting to note that the hypothesis \(n\propto W\) can be proven on real-life data using job advertisement and population data. This is also done in the Results section.

Assuming that jobseekers are willing to accept jobs (or they are aware of the jobs) only with a probability q, the above presented simple argument can be generalized [9] (radiation model with selection (RMwS)), leading to a result with two fitting parameters (\(q,\mu\)):

$$ P_{>}(W)_{\mathrm{RMwS}}=\frac{1-(1-q)^{\mu W+1}}{(\mu W+1) q}. $$ (7)

The travel cost optimized radiation model (TCORM) takes into account the fact that travel costs are distance dependent so in addition with the transited jobs the travel distance, r has to be considered when applying the arguments used in the radiation model. Assuming an exponential distribution kernel for the income distribution, and repeating the arguments from the original radiation model [11] one arrives again to a result with two fitting parameters:

$$ P_{>}(W)_{\mathrm{TCORM}}=\frac{1+\lambda\sqrt{W}}{\mu W+1}. $$ (8)

The λ fitting parameter incorporates both the value of μ, the value of a proportionality constant between the travelled distance and cost of travel and a third constant governing the shape of an assumed exponential-type income distribution [11].

Here we introduce yet another model, offering another one-parameter alternative for the simple RM model. Our alternative, dynamical approach is based on simple master equation for the \(\rho (n,t)=-dP_{>}(n,t)/dn\) probability density and reproduces as a specific case the results of the RM model. We name the model as Flow and Jump Model (FJM). Following the assumptions of the recently introduced growth and reset type models (for a review please consult [18]) we assume now an inverse process: a backward probability flow supplemented by a jump process from the origin to any state with a given n value. The discrete version of the process is depicted in Fig. 1. The continuous master equation has the form:

$$ \frac{d \rho(n,t)}{dt}= \frac{\partial(\eta(n) \rho(n,t))}{\partial n}+ \bigl[\gamma(n)\rho(n,t) \bigr]\rho(0,t). $$ (9)

The above master equation describes a process where there is a local net probability density flow from each state towards the \(n=0\) state and a jump probability from the origin \((n=0)\) to an n state. For the state dependent \(\eta(n)\) and \(\gamma(n)\) rates we consider now simple kernels which makes sense for the commuting process. Definitely the transitions \(0\rightarrow n\) governed by the \(\gamma(n)\rho(n,t)\) rates describes the probability that workers choose a commuting job. \(\gamma(n)\) should decrease with distance (or correspondingly with n) and the proportionality with \(\rho(n,t)\) suggests that where are already many commuters there should also be many good jobs, so it is attractive to commuters.

Figure 1 Dynamics for the FJM model. Sketch of the dynamics that leads in the continuum limit the master equation (9) Full size image

For more details about such dynamical equation, their stability and stationarity please consult [18]. As shown in [18], the stationary solution (\(d\rho _{s}(n,t)/dt=0\)) of (9) is:

$$ \rho_{s}(n)=\frac{\eta(0) \rho_{s}(0)}{\eta(n)}e^{-\int_{0}^{n} \frac{\gamma (x)\rho_{s}(0)}{\eta(x)}\,dx}. $$ (10)

The \(\rho_{s}(0)\) value is obtained from the normalization condition:

$$ \int_{0}^{\infty}\rho_{s}(n) \,dn=1. $$ (11)

For \(\eta(n)\) and \(\gamma(n)\) rates we consider now the simplest kernels which makes sense for the commuting process. For \(\gamma(n)\rho _{s}(0)\) the simplest choice that avoids also the divergence in \(n=0\) is an inverse proportionality:

$$ \gamma(n)\rho_{s}(0)=\frac{C}{n+1}. $$ (12)

C is a constant which fixes also the time unit in the dynamical equation (9). The backward flow characterizes the tendency of the commuters to search for appropriate jobs that are closer to their living places, accepting with a bigger probability jobs that will approach them to their home. This net flow is described by the \(\eta(n)\) terms. The simplest choice that leads to a final equilibrium distribution is:

$$ \eta(n)=\eta=\mathrm{Const}. $$ (13)

For the above \(\gamma(n)\) and \(\eta(n)\) kernels (equations (12) and (13), respectively), and assuming \(a=C/\eta>1\) the solution (10) writes as

$$ \rho_{s}(n)=(a-1) (n+1)^{-a}, $$ (14)

which is a scaling Tsallis–Pareto (or Lomax) type distribution [19]. This probability density leads to the \(P_{>}(n,t)\) probability:

$$ P_{>}(n,t)= \int_{n}^{\infty} \rho_{s}(x) \,dx=(1+n)^{(1-a)}. $$ (15)

With the assumption \(n(r)=\mu W(r)\) we get a slightly modified expectation for \(P_{>}(W)\)

$$ P_{>}(W)_{\mathrm{FJM}}=\frac{1}{(\mu W+1)^{(a-1)}}. $$ (16)

In the followings we demonstrate on real commuting data that the FJM model with the universal choice \(a=7/4\) offers a much improved fit for the real commuting data. For the specific case \(a=2\) one gets back the original radiation model. In principle the model is a two-parameter one, however if we admit the universality of a it becomes similarly with RM a one-parameter model.