diff --git a/Project.toml b/Project.toml
index fe61b716..0aae9b09 100644
--- a/Project.toml
+++ b/Project.toml
@@ -37,7 +37,7 @@ PoissonRandom = "0.4"
 RandomNumbers = "1.5"
 RecursiveArrayTools = "3.12"
 Reexport = "1.0"
-SciMLBase = "2.30.1"
+SciMLBase = "2.46"
 StaticArrays = "1.9"
 SymbolicIndexingInterface = "0.3.13"
 UnPack = "1.0.2"
diff --git a/src/JumpProcesses.jl b/src/JumpProcesses.jl
index f511208c..e72b632d 100644
--- a/src/JumpProcesses.jl
+++ b/src/JumpProcesses.jl
@@ -7,7 +7,7 @@ using RandomNumbers, LinearAlgebra, Markdown, DocStringExtensions
 using DataStructures, PoissonRandom, Random, ArrayInterface
 using FunctionWrappers, UnPack
 using Graphs
-using SciMLBase: SciMLBase
+using SciMLBase: SciMLBase, isdenseplot
 using Base.FastMath: add_fast
 
 import DiffEqBase: DiscreteCallback, init, solve, solve!, plot_indices, initialize!
diff --git a/src/SSA_stepper.jl b/src/SSA_stepper.jl
index f118d132..a9fb7b76 100644
--- a/src/SSA_stepper.jl
+++ b/src/SSA_stepper.jl
@@ -183,11 +183,11 @@ function DiffEqBase.__init(jump_prob::JumpProblem,
         u = typeof(prob.u0)[]
     end
 
-    sol = DiffEqBase.build_solution(prob, alg, t, u, dense = false,
+    save_everystep = any(cb.save_positions)
+    sol = DiffEqBase.build_solution(prob, alg, t, u, dense = save_everystep,
         calculate_error = false,
         stats = DiffEqBase.Stats(0),
         interp = DiffEqBase.ConstantInterpolation(t, u))
-    save_everystep = any(cb.save_positions)
 
     if saveat isa Number
         _saveat = prob.tspan[1]:saveat:prob.tspan[2]
@@ -331,3 +331,9 @@ function DiffEqBase.terminate!(integrator::SSAIntegrator, retcode = ReturnCode.T
 end
 
 export SSAStepper
+
+function SciMLBase.isdenseplot(sol::ODESolution{
+        T, N, uType, uType2, DType, tType, rateType, discType, P,
+        SSAStepper}) where {T, N, uType, uType2, DType, tType, rateType, discType, P}
+    sol.dense
+end
diff --git a/test/save_positions.jl b/test/save_positions.jl
index 1d5e5982..1e5ddc40 100644
--- a/test/save_positions.jl
+++ b/test/save_positions.jl
@@ -1,28 +1,62 @@
-using JumpProcesses, OrdinaryDiffEq, Test
+using JumpProcesses, OrdinaryDiffEq, Test, SciMLBase
 using StableRNGs
 rng = StableRNG(12345)
 
 # test that we only save when a jump occurs
-for alg in (Coevolve(),)
-    u0 = [0]
-    tspan = (0.0, 30.0)
+let
+    for alg in (Coevolve(),)
+        u0 = [0]
+        tspan = (0.0, 30.0)
 
+        dprob = DiscreteProblem(u0, tspan)
+        # set the rate to 0, so that no jump ever occurs; but urate is positive so
+        # Coevolve will consider many candidates before the end of the simmulation.
+        # None of these points should be saved.
+        jump = VariableRateJump((u, p, t) -> 0, (integrator) -> integrator.u[1] += 1;
+            urate = (u, p, t) -> 1.0, rateinterval = (u, p, t) -> 5.0)
+        jumpproblem = JumpProblem(dprob, alg, jump; dep_graph = [[1]],
+            save_positions = (false, true), rng)
+        sol = solve(jumpproblem, SSAStepper())
+        @test sol.t == [0.0, 30.0]
+
+        oprob = ODEProblem((du, u, p, t) -> 0, u0, tspan)
+        jump = VariableRateJump((u, p, t) -> 0, (integrator) -> integrator.u[1] += 1;
+            urate = (u, p, t) -> 1.0, rateinterval = (u, p, t) -> 5.0)
+        jumpproblem = JumpProblem(oprob, alg, jump; dep_graph = [[1]],
+            save_positions = (false, true), rng)
+        sol = solve(jumpproblem, Tsit5(); save_everystep = false)
+        @test sol.t == [0.0, 30.0]
+    end
+end
+
+# test isdenseplot gives correct values for SSAStepper and non-SSAStepper models
+let
+    rate(u, p, t) = max(u[1], 0.0)
+    affect!(integ) = (integ.u[1] -= 1; nothing)
+    crj = ConstantRateJump(rate, affect!)
+    u0 = [10.0]
+    tspan = (0.0, 10.0)
     dprob = DiscreteProblem(u0, tspan)
-    # set the rate to 0, so that no jump ever occurs; but urate is positive so
-    # Coevolve will consider many candidates before the end of the simmulation.
-    # None of these points should be saved.
-    jump = VariableRateJump((u, p, t) -> 0, (integrator) -> integrator.u[1] += 1;
-        urate = (u, p, t) -> 1.0, rateinterval = (u, p, t) -> 5.0)
-    jumpproblem = JumpProblem(dprob, alg, jump; dep_graph = [[1]],
-        save_positions = (false, true))
-    sol = solve(jumpproblem, SSAStepper())
-    @test sol.t == [0.0, 30.0]
+    sps = ((true, true), (true, false), (false, true), (false, false))
+
+    # for pure jump problems dense = save_everystep
+    vals = (true, true, true, false)
+    for (sp, val) in zip(sps, vals)
+        jprob = JumpProblem(dprob, Direct(), crj; save_positions = sp, rng)
+        sol = solve(jprob, SSAStepper())
+        @test SciMLBase.isdenseplot(sol) == val
+    end
+
+    # for mixed problems sol.dense currently ignores save_positions
+    oprob = ODEProblem((du, u, p, t) -> du[1] = 0.1, u0, tspan)
+    for sp in sps
+        jprob = JumpProblem(oprob, Direct(), crj; save_positions = sp, rng)
+        sol = solve(jprob, Tsit5())
+        @test sol.dense == true
+        @test SciMLBase.isdenseplot(sol) == true
 
-    oprob = ODEProblem((du, u, p, t) -> 0, u0, tspan)
-    jump = VariableRateJump((u, p, t) -> 0, (integrator) -> integrator.u[1] += 1;
-        urate = (u, p, t) -> 1.0, rateinterval = (u, p, t) -> 5.0)
-    jumpproblem = JumpProblem(oprob, alg, jump; dep_graph = [[1]],
-        save_positions = (false, true))
-    sol = solve(jumpproblem, Tsit5(); save_everystep = false)
-    @test sol.t == [0.0, 30.0]
+        sol = solve(jprob, Tsit5(); dense = false)
+        @test sol.dense == false
+        @test SciMLBase.isdenseplot(sol) == false
+    end
 end