New climate velocity algorithm is nearly equivalent to simple species distribution modeling methods

In a recent study, Hamann et al. (2015) proposed a new algorithm for computing climatic velocity. The advantage of the new method in comparison with the classical one (Loaire et al., 2009) was that it could effectively avoid infinite velocity and present scale-invariant property. Here, I showed that this novel method actually was a hybrid of two simple methods in species distribution modeling (SDMs). My finding could fundamentally explain why there was a close link between climate velocity and species' potential suitable range as implicitly found in previous works (Burrows et al., 2014). where VA(N) was the velocity in site A using the whole area N as the searching background. Time|future-current| was the year number by subtracting the future-time year to the current-time year. I(●) was the indicator function and returned 1 when the condition inside the parenthesis was satisfied (otherwise returned 0). dist(AB) was the geographic distance between site A and B. d(AcurrentBfuture) was the environmental distance between sites A and B at current and future time (Acurrent and Bfuture), respectively [the same for d(AfutureBcurrent)]. In Hamann et al. (2015)'s study, the climatic distance was measured as the absolute difference of current and future climatic values in site A and B. However, this environmental distance d(●) could be generalized to be any distance metrics. Such a generalization could be used to handle multiple climatic variables simultaneously without performing dimensional reduction (which would lose information). The numerator was to search the minimal distance between sites A and B across the whole area N. Equation 4 actually was SDM methods with a combination of geographic and DOMAIN profile models: both computed the environmental and/or geographic distances for a focused site to the sites occupied by the species. The mathematical formulation of both models was shown as below for detailed comparison. For a site A inside N, its distance (or the unsuitability index of the site for the species) in the past/future time to the sites inside species' current range T could be calculated as: Here, d(●) in the DOMAIN model was the Gower's distance (Gower, 1971). The corresponding suitability index of site A for the species to occupy in the past/future time was , respectively (Carpenter et al., 1993; Hijmans & Elith, 2013). By comparing simple SDM models (5) and (6) to velocity core Eqn 4, MA(T)forward and MA(T)backward could be interpreted as distance metrics measuring the unsuitability of the site A in the past and future time, respectively, for a species to occur: they compared past- and future-climatic conditions of site A, respectively, to current climatic conditions of all the sites within the currently observed range T of the species using both geographic and environmental distances. As such, both MA(T)forward and MA(T)backward were a simple hybrid of geographic and DOMAIN models in doing SDMs: For a test site A, the new velocity algorithm was to (i) use DOMAIN model to search for candidate sites in species' current range T with analogous climate under the constraint of threshold t; (ii) then use geometric model to compute the minimal geographic distance from the test site A to all candidate sites identified at the previous step. Moreover, if t → ∞, . And if species' current range was large enough (T → N), MA(T)forward/backward → MA(N)forward/backward. Therefore, if t → ∞ and N → T, MA(N)forward/backward = VA(N)forward/backward × Time → GA(T). Conclusively, climate velocity algorithm VA(N)forward/backward could be transformed to distance-based SDM models (especially the geographic model) for predicting species' suitable range in the past and future when two constraints were relaxed (t → ∞ and N → T). The author thanked David Roberts for discussions and the editors for comments. This work was supported by the China Scholarship Council (No. 201308180004).

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